Outline. 1 Confidence Intervals for Proportions. 2 Sample Sizes for Proportions. 3 Student s t-distribution. 4 Confidence Intervals without σ

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Outline. 1 Confidence Intervals for Proportions. 2 Sample Sizes for Proportions. 3 Student s t-distribution. 4 Confidence Intervals without σ"

Transcription

1 Outline 1 Confidence Intervals for Proportions 2 Sample Sizes for Proportions 3 Student s t-distribution 4 Confidence Intervals without σ

2 Outline 1 Confidence Intervals for Proportions 2 Sample Sizes for Proportions 3 Student s t-distribution 4 Confidence Intervals without σ

3 Confidence Interval for µ (pretending we know σ) Suppose a population has standard deviation σ. Taking a sample of n individuals, you obtain a sample mean x. Then you can be y-confident that the true mean µ is in the interval (x z σ n, x + z σ n ), where z was a number got from the z-table (using y). That s great for numerical data, but what about categorical data? Question Suppose you take a sample of n individuals from a population and find that x of them are successes, so that your population proportion is p = x n. Then p is our estimate of p, but what is (say) a 95% confidence interval for p?

4 Confidence Interval for µ (pretending we know σ) Suppose a population has standard deviation σ. Taking a sample of n individuals, you obtain a sample mean x. Then you can be y-confident that the true mean µ is in the interval (x z σ n, x + z σ n ), where z was a number got from the z-table (using y). That s great for numerical data, but what about categorical data? Question Suppose you take a sample of n individuals from a population and find that x of them are successes, so that your population proportion is p = x n. Then p is our estimate of p, but what is (say) a 95% confidence interval for p?

5 Some correspondences This... corresponds to... µ p (parameter) x p (statistic) σ n pq n (standard error) (Recall that q = 1 p.)

6 Some correspondences This... corresponds to... µ p (parameter) x p (statistic) σ n pq n (standard error) (Recall that q = 1 p.)

7 Some correspondences This... corresponds to... µ p (parameter) x p (statistic) σ n pq n (standard error) (Recall that q = 1 p.) Answer So let s just make those substitutions in our formula! ) µ is somewhere in (x z n σ, x + z n σ

8 Some correspondences This... corresponds to... µ p (parameter) x p (statistic) σ n pq n (standard error) (Recall that q = 1 p.) Answer So let s just make those substitutions in our formula! ) p is somewhere in (x z n σ, x + z n σ

9 Some correspondences This... corresponds to... µ p (parameter) x p (statistic) σ n pq n (standard error) (Recall that q = 1 p.) Answer So let s just make those substitutions in our formula! ) p is somewhere in ( p z n σ, p + z n σ

10 Some correspondences This... corresponds to... µ p (parameter) x p (statistic) σ n pq n (standard error) (Recall that q = 1 p.) Answer So let s just make those substitutions ( in our formula! ) p is somewhere in p z pq n, p + z pq n

11 Some correspondences This... corresponds to... µ p (parameter) x p (statistic) σ n pq n (standard error) (Recall that q = 1 p.) Answer So let s just make those substitutions ( in our formula! ) p is somewhere in p z pq n, p + z pq n But wait! We don t know p that s the whole point! Happily, it s good enough to use p for p and q = 1 p for q.

12 Some correspondences This... corresponds to... µ p (parameter) x p (statistic) σ n pq n (standard error) (Recall that q = 1 p.) Answer So let s just make those substitutions ( in our formula! ) p is somewhere in p z pq n, p + z pq n But wait! We don t know p that s the whole point! Happily, it s good enough to use p for p and q = 1 p for q.

13 Some correspondences This... corresponds to... µ p (parameter) x p (statistic) σ n pq n (standard error) (Recall that q = 1 p.) Answer So let s just make those substitutions ( in our formula! ) p is somewhere in p z p q n, p + z p q n But wait! We don t know p that s the whole point! Happily, it s good enough to use p for p and q = 1 p for q.

14 Confidence Interval for Proportions If a sample of size n reveals a sample proportion of p, then the confidence interval for the population proportion p is ( ) p q p q p z n, p + z, n where z is the z-score gotten from the confidence level in the usual way. This is good enough as long as the sample size is fairly large, and the population proportion is not too close to 0 or to 1.

15 Confidence Interval for Proportions If a sample of size n reveals a sample proportion of p, then the confidence interval for the population proportion p is ( ) p q p q p z n, p + z, n where z is the z-score gotten from the confidence level in the usual way. This is good enough as long as the sample size is fairly large, and the population proportion is not too close to 0 or to 1.

16 Example: Fish Example 400 randomly chosen people were asked whether they like fish; 160 said yes. Find a 97% confidence interval for p, the proportion of people in the whole population who like fish.

17 Example: Fish Example 400 randomly chosen people were asked whether they like fish; 160 said yes. Find a 97% confidence interval for p, the proportion of people in the whole population who like fish. Solution First, let s see what a 97% confidence interval looks like.

18 Example: Fish We need a 97% confidence interval. 1 Draw the standard normal curve Z. 2 Draw vertical bars and label the middle with That means the remaining area is = That means the left tail has area = The z-table (backwards) tells us the tail ends at So we need 2.17 standard errors!

19 Example: Fish We need a 97% confidence interval. 1 Draw the standard normal curve Z. 2 Draw vertical bars and label the middle with That means the remaining area is = That means the left tail has area = The z-table (backwards) tells us the tail ends at So we need 2.17 standard errors!

20 Example: Fish We need a 97% confidence interval. 1 Draw the standard normal curve Z Draw vertical bars and label the middle with That means the remaining area is = That means the left tail has area = The z-table (backwards) tells us the tail ends at So we need 2.17 standard errors!

21 Example: Fish We need a 97% confidence interval. 1 Draw the standard normal curve Z Draw vertical bars and label the middle with That means the remaining area is = That means the left tail has area = The z-table (backwards) tells us the tail ends at So we need 2.17 standard errors!

22 Example: Fish We need a 97% confidence interval. 1 Draw the standard normal curve Z Draw vertical bars and label the middle with That means the remaining area is = That means the left tail has area = The z-table (backwards) tells us the tail ends at So we need 2.17 standard errors!

23 Example: Fish We need a 97% confidence interval. 1 Draw the standard normal curve Z Draw vertical bars and label the middle with That means the remaining area is = That means the left tail has area = The z-table (backwards) tells us the tail ends at So we need 2.17 standard errors!

24 Example: Fish We need a 97% confidence interval. 1 Draw the standard normal curve Z Draw vertical bars and label the middle with That means the remaining area is = That means the left tail has area = The z-table (backwards) tells us the tail ends at So we need 2.17 standard errors!

25 Example Example: Fish 400 randomly chosen people were asked whether they like fish; 160 said yes. Find a 97% confidence interval for p, the proportion of people in the whole population who like fish. Solution So for 97% confidence we need 2.17 standard errors. Now p = = 0.4, so q = 0.6; also, n = 400. Thus the standard error is p q n = (0.4)(0.6) 400 = Hence our confidence interval is ( ) p q p 2.17 n, p q p n = ( (0.0245), (0.0245)) = (0.347, 0.453) Thus we can be 97% confident that the true proportion of people who like fish is somewhere between 34.7% and 45.3%.

26 Example: Fish Example 400 randomly chosen people were asked whether they like fish; 160 said yes. Find a 97% confidence interval for p, the proportion of people in the whole population who like fish. Solution So for 97% confidence we need 2.17 standard errors. Now p = = 0.4, so q = 0.6; also, n = 400. Thus the standard error is p q n = (0.4)(0.6) 400 = Hence our confidence interval is ( ) p q p 2.17 n, p q p n = ( (0.0245), (0.0245)) = (0.347, 0.453) Thus we can be 97% confident that the true proportion of people who like fish is somewhere between 34.7% and 45.3%.

27 Example Example: Fish 400 randomly chosen people were asked whether they like fish; 160 said yes. Find a 97% confidence interval for p, the proportion of people in the whole population who like fish. Solution So for 97% confidence we need 2.17 standard errors. Now p = = 0.4, so q = 0.6; also, n = 400. Thus the standard error is p q n = (0.4)(0.6) 400 = Hence our confidence interval is ( ) p q p 2.17 n, p q p n = ( (0.0245), (0.0245)) = (0.347, 0.453) Thus we can be 97% confident that the true proportion of people who like fish is somewhere between 34.7% and 45.3%.

28 Example Example: Fish 400 randomly chosen people were asked whether they like fish; 160 said yes. Find a 97% confidence interval for p, the proportion of people in the whole population who like fish. Solution So for 97% confidence we need 2.17 standard errors. Now p = = 0.4, so q = 0.6; also, n = 400. Thus the standard error is p q n = (0.4)(0.6) 400 = Hence our confidence interval is ( ) p q p 2.17 n, p q p n = ( (0.0245), (0.0245)) = (0.347, 0.453) Thus we can be 97% confident that the true proportion of people who like fish is somewhere between 34.7% and 45.3%.

29 Example Example: Fish 400 randomly chosen people were asked whether they like fish; 160 said yes. Find a 97% confidence interval for p, the proportion of people in the whole population who like fish. Solution So for 97% confidence we need 2.17 standard errors. Now p = = 0.4, so q = 0.6; also, n = 400. Thus the standard error is p q n = (0.4)(0.6) 400 = Hence our confidence interval is ( ) p q p 2.17 n, p q p n = ( (0.0245), (0.0245)) = (0.347, 0.453) Thus we can be 97% confident that the true proportion of people who like fish is somewhere between 34.7% and 45.3%.

30 Example Example: Fish 400 randomly chosen people were asked whether they like fish; 160 said yes. Find a 97% confidence interval for p, the proportion of people in the whole population who like fish. Solution So for 97% confidence we need 2.17 standard errors. Now p = = 0.4, so q = 0.6; also, n = 400. Thus the standard error is p q n = (0.4)(0.6) 400 = Hence our confidence interval is ( ) p q p 2.17 n, p q p n = ( (0.0245), (0.0245)) = (0.347, 0.453) Thus we can be 97% confident that the true proportion of people who like fish is somewhere between 34.7% and 45.3%.

31 Example Example: Fish 400 randomly chosen people were asked whether they like fish; 160 said yes. Find a 97% confidence interval for p, the proportion of people in the whole population who like fish. Solution So for 97% confidence we need 2.17 standard errors. Now p = = 0.4, so q = 0.6; also, n = 400. Thus the standard error is p q n = (0.4)(0.6) 400 = Hence our confidence interval is ( ) p q p 2.17 n, p q p n = ( (0.0245), (0.0245)) = (0.347, 0.453) Thus we can be 97% confident that the true proportion of people who like fish is somewhere between 34.7% and 45.3%.

32 Outline 1 Confidence Intervals for Proportions 2 Sample Sizes for Proportions 3 Student s t-distribution 4 Confidence Intervals without σ

33 Finding a good sample size for proportions Last time, we saw how to find the sample size you need to get a confidence interval of a certain size. Can we do that for proportions as well? Example You want to find the true proportion of red-haired people in North Dakota. How many North Dakota residents should you choose randomly in order to be 96% confident that your conclusions are accurate within 3 percentage points?

34 Finding a good sample size for proportions Last time, we saw how to find the sample size you need to get a confidence interval of a certain size. Can we do that for proportions as well? Example You want to find the true proportion of red-haired people in North Dakota. How many North Dakota residents should you choose randomly in order to be 96% confident that your conclusions are accurate within 3 percentage points?

35 Finding a good sample size for proportions Last time, we saw how to find the sample size you need to get a confidence interval of a certain size. Can we do that for proportions as well? Example You want to find the true proportion of red-haired people in North Dakota. How many North Dakota residents should you choose randomly in order to be 96% confident that your conclusions are accurate within 3 percentage points? Solution First, let s see what a 96% confidence interval looks like.

36 Example: Red Hair We need a 96% confidence interval. 1 Draw the standard normal curve Z. 2 Draw vertical bars and label the middle with That means the remaining area is = That means the left tail has area = The z-table (backwards) tells us the tail ends at So we need 2.05 standard errors!

37 Example: Red Hair We need a 96% confidence interval. 1 Draw the standard normal curve Z. 2 Draw vertical bars and label the middle with That means the remaining area is = That means the left tail has area = The z-table (backwards) tells us the tail ends at So we need 2.05 standard errors!

38 Example: Red Hair We need a 96% confidence interval. 1 Draw the standard normal curve Z Draw vertical bars and label the middle with That means the remaining area is = That means the left tail has area = The z-table (backwards) tells us the tail ends at So we need 2.05 standard errors!

39 Example: Red Hair We need a 96% confidence interval. 1 Draw the standard normal curve Z Draw vertical bars and label the middle with That means the remaining area is = That means the left tail has area = The z-table (backwards) tells us the tail ends at So we need 2.05 standard errors!

40 Example: Red Hair We need a 96% confidence interval. 1 Draw the standard normal curve Z Draw vertical bars and label the middle with That means the remaining area is = That means the left tail has area = The z-table (backwards) tells us the tail ends at So we need 2.05 standard errors!

41 Example: Red Hair We need a 96% confidence interval. 1 Draw the standard normal curve Z Draw vertical bars and label the middle with That means the remaining area is = That means the left tail has area = The z-table (backwards) tells us the tail ends at So we need 2.05 standard errors!

42 Example: Red Hair We need a 96% confidence interval. 1 Draw the standard normal curve Z Draw vertical bars and label the middle with That means the remaining area is = That means the left tail has area = The z-table (backwards) tells us the tail ends at So we need 2.05 standard errors!

43 Example Example: Red Hair You want to find the true proportion of red-haired people in North Dakota. How many North Dakota residents should you choose randomly in order to be 96% confident that your conclusions are accurate within 3 percentage points? Solution, cont. The accuracy of our 96% confidence interval is thus 2.05 standard errors, or 2.05 p q n within 0.03, so we want We want that accuracy to be p q n. Solving for n, 0.03 n 2.05 p q n 2.05 p q 0.03 = p q ( 2 n = p q. We need n to be at least p q. But we don t know p and q until we do the survey!

44 Example: Red Hair Example You want to find the true proportion of red-haired people in North Dakota. How many North Dakota residents should you choose randomly in order to be 96% confident that your conclusions are accurate within 3 percentage points? Solution, cont. The accuracy of our 96% confidence interval is thus 2.05 standard errors, or 2.05 p q n within 0.03, so we want We want that accuracy to be p q n. Solving for n, 0.03 n 2.05 p q n 2.05 p q 0.03 = p q ( 2 n = p q. We need n to be at least p q. But we don t know p and q until we do the survey!

45 Example Example: Red Hair You want to find the true proportion of red-haired people in North Dakota. How many North Dakota residents should you choose randomly in order to be 96% confident that your conclusions are accurate within 3 percentage points? Solution, cont. The accuracy of our 96% confidence interval is thus 2.05 standard errors, or 2.05 p q n within 0.03, so we want We want that accuracy to be p q n. Solving for n, 0.03 n 2.05 p q n 2.05 p q 0.03 = p q ( 2 n = p q. We need n to be at least p q. But we don t know p and q until we do the survey!

46 Example Example: Red Hair You want to find the true proportion of red-haired people in North Dakota. How many North Dakota residents should you choose randomly in order to be 96% confident that your conclusions are accurate within 3 percentage points? Solution, cont. The accuracy of our 96% confidence interval is thus 2.05 standard errors, or 2.05 p q n within 0.03, so we want We want that accuracy to be p q n. Solving for n, 0.03 n 2.05 p q n 2.05 p q 0.03 = p q ( 2 n = p q. We need n to be at least p q. But we don t know p and q until we do the survey!

47 Example Example: Red Hair You want to find the true proportion of red-haired people in North Dakota. How many North Dakota residents should you choose randomly in order to be 96% confident that your conclusions are accurate within 3 percentage points? Solution, cont. The accuracy of our 96% confidence interval is thus 2.05 standard errors, or 2.05 p q n within 0.03, so we want We want that accuracy to be p q n. Solving for n, 0.03 n 2.05 p q n 2.05 p q 0.03 = p q ( 2 n = p q. We need n to be at least p q. But we don t know p and q until we do the survey!

48 Example Example: Red Hair You want to find the true proportion of red-haired people in North Dakota. How many North Dakota residents should you choose randomly in order to be 96% confident that your conclusions are accurate within 3 percentage points? Solution, cont. The accuracy of our 96% confidence interval is thus 2.05 standard errors, or 2.05 p q n within 0.03, so we want We want that accuracy to be p q n. Solving for n, 0.03 n 2.05 p q n 2.05 p q 0.03 = p q ( 2 n = p q. We need n to be at least p q. But we don t know p and q until we do the survey!

49 Example Example: Red Hair You want to find the true proportion of red-haired people in North Dakota. How many North Dakota residents should you choose randomly in order to be 96% confident that your conclusions are accurate within 3 percentage points? Solution, cont. The accuracy of our 96% confidence interval is thus 2.05 standard errors, or 2.05 p q n within 0.03, so we want We want that accuracy to be p q n. Solving for n, 0.03 n 2.05 p q n 2.05 p q 0.03 = p q ( 2 n = p q. We need n to be at least p q. But we don t know p and q until we do the survey!

50 Example Example: Red Hair You want to find the true proportion of red-haired people in North Dakota. How many North Dakota residents should you choose randomly in order to be 96% confident that your conclusions are accurate within 3 percentage points? Solution, cont. The accuracy of our 96% confidence interval is thus 2.05 standard errors, or 2.05 p q n within 0.03, so we want We want that accuracy to be p q n. Solving for n, 0.03 n 2.05 p q n 2.05 p q 0.03 = p q ( 2 n = p q. We need n to be at least p q. But we don t know p and q until we do the survey!

51 Example Example: Red Hair You want to find the true proportion of red-haired people in North Dakota. How many North Dakota residents should you choose randomly in order to be 96% confident that your conclusions are accurate within 3 percentage points? Solution, cont. The accuracy of our 96% confidence interval is thus 2.05 standard errors, or 2.05 p q n within 0.03, so we want We want that accuracy to be p q n. Solving for n, 0.03 n 2.05 p q n 2.05 p q 0.03 = p q ( 2 n = p q. We need n to be at least p q. But we don t know p and q until we do the survey!

52 What saves us p(1 p) 1 p Fortunately, we know p is somewhere between 0 and 1. Also, p q = p(1 p). If we graph the function p(1 p), we see that it can t get too large! In fact, the largest it can be is That is, p q 0.25.

53 What saves us p(1 p) 1 p Fortunately, we know p is somewhere between 0 and 1. Also, p q = p(1 p). If we graph the function p(1 p), we see that it can t get too large! In fact, the largest it can be is That is, p q 0.25.

54 What saves us p(1 p) p Fortunately, we know p is somewhere between 0 and 1. Also, p q = p(1 p). If we graph the function p(1 p), we see that it can t get too large! In fact, the largest it can be is That is, p q 0.25.

55 What saves us p(1 p) p Fortunately, we know p is somewhere between 0 and 1. Also, p q = p(1 p). If we graph the function p(1 p), we see that it can t get too large! In fact, the largest it can be is That is, p q 0.25.

56 Example: Red Hair Example You want to find the true proportion of red-haired people in North Dakota. How many North Dakota residents should you choose randomly in order to be 96% confident that your conclusions are accurate within 3 percentage points? Solution, cont. We found out that we need n to be at least p q. Since the largest p q can be is 0.25, that means we need n = Thus we need at least 1,168 people in our survey in order to be 96% sure that our survey is accurate within three percentage points.

57 Example: Red Hair Example You want to find the true proportion of red-haired people in North Dakota. How many North Dakota residents should you choose randomly in order to be 96% confident that your conclusions are accurate within 3 percentage points? Solution, cont. We found out that we need n to be at least p q. Since the largest p q can be is 0.25, that means we need n = Thus we need at least 1,168 people in our survey in order to be 96% sure that our survey is accurate within three percentage points.

58 Example: Red Hair Example You want to find the true proportion of red-haired people in North Dakota. How many North Dakota residents should you choose randomly in order to be 96% confident that your conclusions are accurate within 3 percentage points? Solution, cont. We found out that we need n to be at least p q. Since the largest p q can be is 0.25, that means we need n = Thus we need at least 1,168 people in our survey in order to be 96% sure that our survey is accurate within three percentage points.

59 Summary Finding sample size for proportions 1 Find out what z-score you need for the desired confidence level. 2 Set your desired accuracy equal to z p q n. 3 Plug in the z-score you found. 4 Instead of p q, use their maximum value, namely Now solve for n.

60 Outline 1 Confidence Intervals for Proportions 2 Sample Sizes for Proportions 3 Student s t-distribution 4 Confidence Intervals without σ

61 Getting rid of σ The 95% confidence interval is (x 2 σ n, x + 2 σ n ) In practice, we know x and n, but we don t know σ. What can we do? Our best guess for σ is s, the sample standard deviation.

62 Sample Variance and Standard Deviation Definition If our sample yields the list of numbers {x 1, x 2,..., x n }, then the sample variance is given by s 2 = (x 1 x) 2 + (x 2 x) (x n x) 2. n 1 The sample standard deviation s is the square root of the sample variance. Alternate form An easier version for computing the sample variance is s 2 = (x 1) 2 + (x 2 ) (x n ) 2 nx 2. n 1

63 Using s instead of σ The simplest thing to do would be to use s instead of σ in our confidence interval formula. We could try (x z σ n, x + z σ n ). This actually works surprisingly well... For the rest of the time, we need another approach, known as Student s t-distribution.

64 Using s instead of σ The simplest thing to do would be to use s instead of σ in our confidence interval formula. We could try (x z σ n, x + z σ n ). This actually works surprisingly well... For the rest of the time, we need another approach, known as Student s t-distribution.

65 Using s instead of σ The simplest thing to do would be to use s instead of σ in our confidence interval formula. We could try ( x z s n, x + z s n ). This actually works surprisingly well... For the rest of the time, we need another approach, known as Student s t-distribution.

66 Using s instead of σ The simplest thing to do would be to use s instead of σ in our confidence interval formula. We could try ( x z s n, x + z s n ). This actually works surprisingly well... For the rest of the time, we need another approach, known as Student s t-distribution.

67 Using s instead of σ The simplest thing to do would be to use s instead of σ in our confidence interval formula. We could try ( x z s n, x + z s n ). This actually works surprisingly well... some of the time. For the rest of the time, we need another approach, known as Student s t-distribution.

68 Using s instead of σ The simplest thing to do would be to use s instead of σ in our confidence interval formula. We could try ( x z s n, x + z s n ). This actually works surprisingly well... some of the time. For the rest of the time, we need another approach, known as Student s t-distribution.

69 Where it began... William S. Gosset Student

70 Where it began... William S. Gosset Student Arthur Guinness Son & Co. Ltd. good

71 Where it began... William S. Gosset Student Arthur Guinness Son & Co. Ltd. good In the early 1900 s, Guinness employed Gosset as a statistician to help improve their beer. Brewing is a long, expensive process, and Gosset often had only a few batches of beer in his samples. Gosset found that using s n worked well when he had a large n, but when n was small, it was producing confidence intervals that were too small.

72 Where it began... William S. Gosset Student Arthur Guinness Son & Co. Ltd. good In the early 1900 s, Guinness employed Gosset as a statistician to help improve their beer. Brewing is a long, expensive process, and Gosset often had only a few batches of beer in his samples. Gosset found that using s n worked well when he had a large n, but when n was small, it was producing confidence intervals that were too small.

73 Where it began... William S. Gosset Student Arthur Guinness Son & Co. Ltd. good In the early 1900 s, Guinness employed Gosset as a statistician to help improve their beer. Brewing is a long, expensive process, and Gosset often had only a few batches of beer in his samples. Gosset found that using s n worked well when he had a large n, but when n was small, it was producing confidence intervals that were too small.

74 Why it goes wrong Before, when we constructed the 95% confidence interval off of µ, we got error bars of 2 σ n from the uncertainty of where µ was. x But if we don t know σ, then that just adds to our uncertainty! x

75 Why it goes wrong Before, when we constructed the 95% confidence interval off of µ, we got error bars of 2 σ n from the uncertainty of where µ was. x 2 σ n x x + 2 σ n But if we don t know σ, then that just adds to our uncertainty! x

76 Why it goes wrong Before, when we constructed the 95% confidence interval off of µ, we got error bars of 2 σ n from the uncertainty of where µ was. uncertainty from unknown µ x 2 σ n x x + 2 σ n But if we don t know σ, then that just adds to our uncertainty! x

77 Why it goes wrong Before, when we constructed the 95% confidence interval off of µ, we got error bars of 2 σ n from the uncertainty of where µ was. uncertainty from unknown µ x 2 σ n x x + 2 σ n But if we don t know σ, then that just adds to our uncertainty! x

78 Why it goes wrong Before, when we constructed the 95% confidence interval off of µ, we got error bars of 2 σ n from the uncertainty of where µ was. uncertainty from unknown µ x 2 σ n x x + 2 σ n But if we don t know σ, then that just adds to our uncertainty! uncertainty from unknown µ x 2 s n x x + 2 s n

79 Why it goes wrong Before, when we constructed the 95% confidence interval off of µ, we got error bars of 2 σ n from the uncertainty of where µ was. uncertainty from unknown µ x 2 σ n x x + 2 σ n But if we don t know σ, then that just adds to our uncertainty! uncertainty from unknown µ extra uncertainty from unknown σ x 2 s n x x + 2 s n

80 Gosset s observation If we use s instead of σ, then we re more uncertain. Therefore we need more s n s than we would need of σ n s. We got the number of standard errors to use from the Z -distribution. So that s the wrong distribution to use!

81 If we were using σ, within 2 standard errors we would have 95% confidence. Because we re working with s instead of σ, we have less confidence! So we need a flatter distribution than Z!

82 95% If we were using σ, within 2 standard errors we would have 95% confidence. Because we re working with s instead of σ, we have less confidence! So we need a flatter distribution than Z!

83 88% If we were using σ, within 2 standard errors we would have 95% confidence. Because we re working with s instead of σ, we have less confidence! So we need a flatter distribution than Z!

84 88% If we were using σ, within 2 standard errors we would have 95% confidence. Because we re working with s instead of σ, we have less confidence! So we need a flatter distribution than Z!

85 The Story of Student Wm. S. Gosset discovered the flatter distribution that gives the confidence intervals with small sample sizes. Some years earlier, a Guinness employee had published some of the company s brewing secrets, so Guinness prohibited its employees from publishing. Gosset pleaded with Guinness to let him publish math. They finally gave him permission, under one condition.

86

87 The t-distribution Gosset found the formula for the right distribution for small samples. There s a different distribution for each sample size. If your sample size is n, you use the t-distribution with n 1 degrees of freedom.

88 z-distribution The t-distribution t-distribution with 1 degrees of freedom Gosset found the formula for the right distribution for small samples. There s a different distribution for each sample size. If your sample size is n, you use the t-distribution with n 1 degrees of freedom.

89 z-distribution The t-distribution t-distribution with 2 degrees of freedom Gosset found the formula for the right distribution for small samples. There s a different distribution for each sample size. If your sample size is n, you use the t-distribution with n 1 degrees of freedom.

90 z-distribution The t-distribution t-distribution with 3 degrees of freedom Gosset found the formula for the right distribution for small samples. There s a different distribution for each sample size. If your sample size is n, you use the t-distribution with n 1 degrees of freedom.

91 z-distribution The t-distribution t-distribution with 4 degrees of freedom Gosset found the formula for the right distribution for small samples. There s a different distribution for each sample size. If your sample size is n, you use the t-distribution with n 1 degrees of freedom.

92 z-distribution The t-distribution t-distribution with 5 degrees of freedom Gosset found the formula for the right distribution for small samples. There s a different distribution for each sample size. If your sample size is n, you use the t-distribution with n 1 degrees of freedom.

93 z-distribution The t-distribution t-distribution with 6 degrees of freedom Gosset found the formula for the right distribution for small samples. There s a different distribution for each sample size. If your sample size is n, you use the t-distribution with n 1 degrees of freedom.

94 z-distribution The t-distribution t-distribution with 7 degrees of freedom Gosset found the formula for the right distribution for small samples. There s a different distribution for each sample size. If your sample size is n, you use the t-distribution with n 1 degrees of freedom.

95 z-distribution The t-distribution t-distribution with 8 degrees of freedom Gosset found the formula for the right distribution for small samples. There s a different distribution for each sample size. If your sample size is n, you use the t-distribution with n 1 degrees of freedom.

96 z-distribution The t-distribution t-distribution with 9 degrees of freedom Gosset found the formula for the right distribution for small samples. There s a different distribution for each sample size. If your sample size is n, you use the t-distribution with n 1 degrees of freedom.

97 z-distribution The t-distribution t-distribution with 10 degrees of freedom Gosset found the formula for the right distribution for small samples. There s a different distribution for each sample size. If your sample size is n, you use the t-distribution with n 1 degrees of freedom.

98 z-distribution The t-distribution t-distribution with 11 degrees of freedom Gosset found the formula for the right distribution for small samples. There s a different distribution for each sample size. If your sample size is n, you use the t-distribution with n 1 degrees of freedom.

99 z-distribution The t-distribution t-distribution with 12 degrees of freedom Gosset found the formula for the right distribution for small samples. There s a different distribution for each sample size. If your sample size is n, you use the t-distribution with n 1 degrees of freedom.

100 z-distribution The t-distribution t-distribution with 13 degrees of freedom Gosset found the formula for the right distribution for small samples. There s a different distribution for each sample size. If your sample size is n, you use the t-distribution with n 1 degrees of freedom.

101 z-distribution The t-distribution t-distribution with 14 degrees of freedom Gosset found the formula for the right distribution for small samples. There s a different distribution for each sample size. If your sample size is n, you use the t-distribution with n 1 degrees of freedom.

102 z-distribution The t-distribution t-distribution with 15 degrees of freedom Gosset found the formula for the right distribution for small samples. There s a different distribution for each sample size. If your sample size is n, you use the t-distribution with n 1 degrees of freedom.

103 z-distribution The t-distribution t-distribution with 16 degrees of freedom Gosset found the formula for the right distribution for small samples. There s a different distribution for each sample size. If your sample size is n, you use the t-distribution with n 1 degrees of freedom.

104 z-distribution The t-distribution t-distribution with 17 degrees of freedom Gosset found the formula for the right distribution for small samples. There s a different distribution for each sample size. If your sample size is n, you use the t-distribution with n 1 degrees of freedom.

105 z-distribution The t-distribution t-distribution with 18 degrees of freedom Gosset found the formula for the right distribution for small samples. There s a different distribution for each sample size. If your sample size is n, you use the t-distribution with n 1 degrees of freedom.

106 z-distribution The t-distribution t-distribution with 19 degrees of freedom Gosset found the formula for the right distribution for small samples. There s a different distribution for each sample size. If your sample size is n, you use the t-distribution with n 1 degrees of freedom.

107 z-distribution The t-distribution t-distribution with 20 degrees of freedom Gosset found the formula for the right distribution for small samples. There s a different distribution for each sample size. If your sample size is n, you use the t-distribution with n 1 degrees of freedom.

108 z-distribution The t-distribution t-distribution with 21 degrees of freedom Gosset found the formula for the right distribution for small samples. There s a different distribution for each sample size. If your sample size is n, you use the t-distribution with n 1 degrees of freedom.

109 z-distribution The t-distribution t-distribution with 22 degrees of freedom Gosset found the formula for the right distribution for small samples. There s a different distribution for each sample size. If your sample size is n, you use the t-distribution with n 1 degrees of freedom.

110 z-distribution The t-distribution t-distribution with 23 degrees of freedom Gosset found the formula for the right distribution for small samples. There s a different distribution for each sample size. If your sample size is n, you use the t-distribution with n 1 degrees of freedom.

111 z-distribution The t-distribution t-distribution with 24 degrees of freedom Gosset found the formula for the right distribution for small samples. There s a different distribution for each sample size. If your sample size is n, you use the t-distribution with n 1 degrees of freedom.

112 z-distribution The t-distribution t-distribution with 25 degrees of freedom Gosset found the formula for the right distribution for small samples. There s a different distribution for each sample size. If your sample size is n, you use the t-distribution with n 1 degrees of freedom.

113 z-distribution The t-distribution t-distribution with 26 degrees of freedom Gosset found the formula for the right distribution for small samples. There s a different distribution for each sample size. If your sample size is n, you use the t-distribution with n 1 degrees of freedom.

114 z-distribution The t-distribution t-distribution with 27 degrees of freedom Gosset found the formula for the right distribution for small samples. There s a different distribution for each sample size. If your sample size is n, you use the t-distribution with n 1 degrees of freedom.

115 z-distribution The t-distribution t-distribution with 28 degrees of freedom Gosset found the formula for the right distribution for small samples. There s a different distribution for each sample size. If your sample size is n, you use the t-distribution with n 1 degrees of freedom.

116 z-distribution The t-distribution t-distribution with 29 degrees of freedom Gosset found the formula for the right distribution for small samples. There s a different distribution for each sample size. If your sample size is n, you use the t-distribution with n 1 degrees of freedom.

117 z-distribution The t-distribution t-distribution with 30 degrees of freedom Gosset found the formula for the right distribution for small samples. There s a different distribution for each sample size. If your sample size is n, you use the t-distribution with n 1 degrees of freedom.

118 z-distribution The t-distribution t-distribution with 30 degrees of freedom Gosset found the formula for the right distribution for small samples. There s a different distribution for each sample size. If your sample size is n, you use the t-distribution with n 1 degrees of freedom. If n 30, then the t-distribution is almost exactly the normal curve Z.

119 Outline 1 Confidence Intervals for Proportions 2 Sample Sizes for Proportions 3 Student s t-distribution 4 Confidence Intervals without σ

120 Student s Conclusions To make a confidence interval when we don t know σ, we replace σ n with our estimate s n. If our sample size n is at least 30, we use the Z -curve just like last time. If our sample size n is less than 30, we use the t-curve for n 1 degrees of freedom. So the only change in our procedure is to look up the numbers in a different table!

121 Finding a y-confidence interval from a small sample 1 Subtract 1 from the sample size n to get n 1 degrees of freedom. 2 Draw Student s t-distribution with n 1 degrees of freedom. 3 Draw two vertical bars symmetrically on the graph, and label the middle with y. 4 That means the remaining area is 1 y. 5 That means the left tail has area 1 y 2. 6 Use the appropriate t-table to learn where that tail ends! 7 Use that many standard errors s n!

122 Finding a y-confidence interval from a small sample 1 Subtract 1 from the sample size n to get n 1 degrees of freedom. 2 Draw Student s t-distribution with n 1 degrees of freedom. 3 Draw two vertical bars symmetrically on the graph, and label the middle with y. 4 That means the remaining area is 1 y. 5 That means the left tail has area 1 y 2. 6 Use the appropriate t-table to learn where that tail ends! 7 Use that many standard errors s n!

123 Finding a y-confidence interval from a small sample 1 Subtract 1 from the sample size n to get n 1 degrees of freedom. 2 Draw Student s t-distribution with n 1 degrees of freedom. y 3 Draw two vertical bars symmetrically on the graph, and label the middle with y. 4 That means the remaining area is 1 y. 5 That means the left tail has area 1 y 2. 6 Use the appropriate t-table to learn where that tail ends! 7 Use that many standard errors s n!

124 Finding a y-confidence interval from a small sample 1 Subtract 1 from the sample size n to get n 1 degrees of freedom. 2 Draw Student s t-distribution with n 1 degrees of freedom. y 1 y 3 Draw two vertical bars symmetrically on the graph, and label the middle with y. 4 That means the remaining area is 1 y. 5 That means the left tail has area 1 y 2. 6 Use the appropriate t-table to learn where that tail ends! 7 Use that many standard errors s n!

125 Finding a y-confidence interval from a small sample 1 Subtract 1 from the sample size n to get n 1 degrees of freedom. 2 Draw Student s t-distribution with n 1 degrees of freedom. 1 y 2 y 1 y 3 Draw two vertical bars symmetrically on the graph, and label the middle with y. 4 That means the remaining area is 1 y. 5 That means the left tail has area 1 y 2. 6 Use the appropriate t-table to learn where that tail ends! 7 Use that many standard errors s n!

126 Finding a y-confidence interval from a small sample 1 Subtract 1 from the sample size n to get n 1 degrees of freedom. 2 Draw Student s t-distribution with n 1 degrees of freedom. 1 y 2 y 1 y t 3 Draw two vertical bars symmetrically on the graph, and label the middle with y. 4 That means the remaining area is 1 y. 5 That means the left tail has area 1 y 2. 6 Use the appropriate t-table to learn where that tail ends! 7 Use that many standard errors s n!

127 Finding a y-confidence interval from a small sample 1 Subtract 1 from the sample size n to get n 1 degrees of freedom. 2 Draw Student s t-distribution with n 1 degrees of freedom. 1 y 2 y 1 y t 3 Draw two vertical bars symmetrically on the graph, and label the middle with y. 4 That means the remaining area is 1 y. 5 That means the left tail has area 1 y 2. 6 Use the appropriate t-table to learn where that tail ends! 7 Use that many standard errors s n!

128 Example Example: Sugar Mrs. Smith is worried about her family s health, so she keeps track of how much sugar they use. In five randomly picked weeks, they used the following amounts of sugar (in pounds): Construct a 94% confidence interval for the true mean µ.

129 Example Example: Sugar Mrs. Smith is worried about her family s health, so she keeps track of how much sugar they use. In five randomly picked weeks, they used the following amounts of sugar (in pounds): Construct a 94% confidence interval for the true mean µ. Solution First we need to find the sample mean x and sample standard deviation s x = = s 2 = = 0.545, 5 1 so s = = Next, we need to see what a 94% confidence interval looks like for a sample size of n = 5.

130 Example Example: Sugar Mrs. Smith is worried about her family s health, so she keeps track of how much sugar they use. In five randomly picked weeks, they used the following amounts of sugar (in pounds): Construct a 94% confidence interval for the true mean µ. Solution First we need to find the sample mean x and sample standard deviation s x = = s 2 = = 0.545, 5 1 so s = = Next, we need to see what a 94% confidence interval looks like for a sample size of n = 5.

131 Example Example: Sugar Mrs. Smith is worried about her family s health, so she keeps track of how much sugar they use. In five randomly picked weeks, they used the following amounts of sugar (in pounds): Construct a 94% confidence interval for the true mean µ. Solution First we need to find the sample mean x and sample standard deviation s x = = s 2 = = 0.545, 5 1 so s = = Next, we need to see what a 94% confidence interval looks like for a sample size of n = 5.

132 Example: Sugar 1 n = 5, so we need 5 1 = 4 degrees of freedom. 2 Draw Student s t-distribution with 4 degrees of freedom. 3 Draw two vertical bars symmetrically on the graph, and label the middle with That means the remaining area is That means the left tail has area The t-table for 4 degrees of freedom says the tail ends at So we need 2.60 standard errors s n!

133 Example: Sugar 1 n = 5, so we need 5 1 = 4 degrees of freedom. 2 Draw Student s t-distribution with 4 degrees of freedom. 3 Draw two vertical bars symmetrically on the graph, and label the middle with That means the remaining area is That means the left tail has area The t-table for 4 degrees of freedom says the tail ends at So we need 2.60 standard errors s n!

134 Example: Sugar 1 n = 5, so we need 5 1 = 4 degrees of freedom. 2 Draw Student s t-distribution with 4 degrees of freedom Draw two vertical bars symmetrically on the graph, and label the middle with That means the remaining area is That means the left tail has area The t-table for 4 degrees of freedom says the tail ends at So we need 2.60 standard errors s n!

135 Example: Sugar 1 n = 5, so we need 5 1 = 4 degrees of freedom. 2 Draw Student s t-distribution with 4 degrees of freedom Draw two vertical bars symmetrically on the graph, and label the middle with That means the remaining area is That means the left tail has area The t-table for 4 degrees of freedom says the tail ends at So we need 2.60 standard errors s n!

136 Example: Sugar 1 n = 5, so we need 5 1 = 4 degrees of freedom. 2 Draw Student s t-distribution with 4 degrees of freedom Draw two vertical bars symmetrically on the graph, and label the middle with That means the remaining area is That means the left tail has area The t-table for 4 degrees of freedom says the tail ends at So we need 2.60 standard errors s n!

137 Example: Sugar 1 n = 5, so we need 5 1 = 4 degrees of freedom. 2 Draw Student s t-distribution with 4 degrees of freedom Draw two vertical bars symmetrically on the graph, and label the middle with That means the remaining area is That means the left tail has area The t-table for 4 degrees of freedom says the tail ends at So we need 2.60 standard errors s n!

138 Example: Sugar 1 n = 5, so we need 5 1 = 4 degrees of freedom. 2 Draw Student s t-distribution with 4 degrees of freedom Draw two vertical bars symmetrically on the graph, and label the middle with That means the remaining area is That means the left tail has area The t-table for 4 degrees of freedom says the tail ends at So we need 2.60 standard errors s n!

139 Example Example: Sugar Mrs. Smith is worried about her family s health, so she keeps track of how much sugar they use. In five randomly picked weeks, they used the following amounts of sugar (in pounds): Construct a 94% confidence interval for the true mean consumption µ. Solution So we need 2.60 standard errors; recall that n = 5, x = 4.6, and s = So the confidence interval is ( x 2.60 s, x s ) n n ( = , ) 5 5 = (3.741, 5.459). Thus Mrs. Smith can be 94% sure that her family averages between pounds and pounds of sugar per week.

140 Example Example: Sugar Mrs. Smith is worried about her family s health, so she keeps track of how much sugar they use. In five randomly picked weeks, they used the following amounts of sugar (in pounds): Construct a 94% confidence interval for the true mean consumption µ. Solution So we need 2.60 standard errors; recall that n = 5, x = 4.6, and s = So the confidence interval is ( x 2.60 s, x s ) n n ( = , ) 5 5 = (3.741, 5.459). Thus Mrs. Smith can be 94% sure that her family averages between pounds and pounds of sugar per week.

141 Example Example: Sugar Mrs. Smith is worried about her family s health, so she keeps track of how much sugar they use. In five randomly picked weeks, they used the following amounts of sugar (in pounds): Construct a 94% confidence interval for the true mean consumption µ. Solution So we need 2.60 standard errors; recall that n = 5, x = 4.6, and s = So the confidence interval is ( x 2.60 s, x s ) n n ( = , ) 5 5 = (3.741, 5.459). Thus Mrs. Smith can be 94% sure that her family averages between pounds and pounds of sugar per week.

142 Example Example: Sugar Mrs. Smith is worried about her family s health, so she keeps track of how much sugar they use. In five randomly picked weeks, they used the following amounts of sugar (in pounds): Construct a 94% confidence interval for the true mean consumption µ. Solution So we need 2.60 standard errors; recall that n = 5, x = 4.6, and s = So the confidence interval is ( x 2.60 s, x s ) n n ( = , ) 5 5 = (3.741, 5.459). Thus Mrs. Smith can be 94% sure that her family averages between pounds and pounds of sugar per week.

AP * Statistics Review

AP * Statistics Review AP * Statistics Review Confidence Intervals Teacher Packet AP* is a trademark of the College Entrance Examination Board. The College Entrance Examination Board was not involved in the production of this

More information

Basic Statistics. Probability and Confidence Intervals

Basic Statistics. Probability and Confidence Intervals Basic Statistics Probability and Confidence Intervals Probability and Confidence Intervals Learning Intentions Today we will understand: Interpreting the meaning of a confidence interval Calculating the

More information

Sampling Central Limit Theorem Proportions. Outline. 1 Sampling. 2 Central Limit Theorem. 3 Proportions

Sampling Central Limit Theorem Proportions. Outline. 1 Sampling. 2 Central Limit Theorem. 3 Proportions Outline 1 Sampling 2 Central Limit Theorem 3 Proportions Outline 1 Sampling 2 Central Limit Theorem 3 Proportions Populations and samples When we use statistics, we are trying to find out information about

More information

4. Introduction to Statistics

4. Introduction to Statistics Statistics for Engineers 4-1 4. Introduction to Statistics Descriptive Statistics Types of data A variate or random variable is a quantity or attribute whose value may vary from one unit of investigation

More information

Standard Deviation Calculator

Standard Deviation Calculator CSS.com Chapter 35 Standard Deviation Calculator Introduction The is a tool to calculate the standard deviation from the data, the standard error, the range, percentiles, the COV, confidence limits, or

More information

An interval estimate (confidence interval) is an interval, or range of values, used to estimate a population parameter. For example 0.476<p<0.

An interval estimate (confidence interval) is an interval, or range of values, used to estimate a population parameter. For example 0.476<p<0. Lecture #7 Chapter 7: Estimates and sample sizes In this chapter, we will learn an important technique of statistical inference to use sample statistics to estimate the value of an unknown population parameter.

More information

Statistical Inference

Statistical Inference Statistical Inference Idea: Estimate parameters of the population distribution using data. How: Use the sampling distribution of sample statistics and methods based on what would happen if we used this

More information

Estimation of the Mean and Proportion

Estimation of the Mean and Proportion 1 Excel Manual Estimation of the Mean and Proportion Chapter 8 While the spreadsheet setups described in this guide may seem to be getting more complicated, once they are created (and tested!), they will

More information

MEASURES OF VARIATION

MEASURES OF VARIATION NORMAL DISTRIBTIONS MEASURES OF VARIATION In statistics, it is important to measure the spread of data. A simple way to measure spread is to find the range. But statisticians want to know if the data are

More information

Objectives. 6.1, 7.1 Estimating with confidence (CIS: Chapter 10) CI)

Objectives. 6.1, 7.1 Estimating with confidence (CIS: Chapter 10) CI) Objectives 6.1, 7.1 Estimating with confidence (CIS: Chapter 10) Statistical confidence (CIS gives a good explanation of a 95% CI) Confidence intervals. Further reading http://onlinestatbook.com/2/estimation/confidence.html

More information

Hypothesis Testing Level I Quantitative Methods. IFT Notes for the CFA exam

Hypothesis Testing Level I Quantitative Methods. IFT Notes for the CFA exam Hypothesis Testing 2014 Level I Quantitative Methods IFT Notes for the CFA exam Contents 1. Introduction... 3 2. Hypothesis Testing... 3 3. Hypothesis Tests Concerning the Mean... 10 4. Hypothesis Tests

More information

Confidence intervals, t tests, P values

Confidence intervals, t tests, P values Confidence intervals, t tests, P values Joe Felsenstein Department of Genome Sciences and Department of Biology Confidence intervals, t tests, P values p.1/31 Normality Everybody believes in the normal

More information

Sampling Distribution of a Sample Proportion

Sampling Distribution of a Sample Proportion Sampling Distribution of a Sample Proportion From earlier material remember that if X is the count of successes in a sample of n trials of a binomial random variable then the proportion of success is given

More information

Margin of Error When Estimating a Population Proportion

Margin of Error When Estimating a Population Proportion Margin of Error When Estimating a Population Proportion Student Outcomes Students use data from a random sample to estimate a population proportion. Students calculate and interpret margin of error in

More information

Outline. Correlation & Regression, III. Review. Relationship between r and regression

Outline. Correlation & Regression, III. Review. Relationship between r and regression Outline Correlation & Regression, III 9.07 4/6/004 Relationship between correlation and regression, along with notes on the correlation coefficient Effect size, and the meaning of r Other kinds of correlation

More information

Lesson 7 Z-Scores and Probability

Lesson 7 Z-Scores and Probability Lesson 7 Z-Scores and Probability Outline Introduction Areas Under the Normal Curve Using the Z-table Converting Z-score to area -area less than z/area greater than z/area between two z-values Converting

More information

Chapter 3 Normal Distribution

Chapter 3 Normal Distribution Chapter 3 Normal Distribution Density curve A density curve is an idealized histogram, a mathematical model; the curve tells you what values the quantity can take and how likely they are. Example Height

More information

Confidence intervals

Confidence intervals Confidence intervals Today, we re going to start talking about confidence intervals. We use confidence intervals as a tool in inferential statistics. What this means is that given some sample statistics,

More information

Hypothesis Testing. Bluman Chapter 8

Hypothesis Testing. Bluman Chapter 8 CHAPTER 8 Learning Objectives C H A P T E R E I G H T Hypothesis Testing 1 Outline 8-1 Steps in Traditional Method 8-2 z Test for a Mean 8-3 t Test for a Mean 8-4 z Test for a Proportion 8-5 2 Test for

More information

Cents and the Central Limit Theorem Overview of Lesson GAISE Components Common Core State Standards for Mathematical Practice

Cents and the Central Limit Theorem Overview of Lesson GAISE Components Common Core State Standards for Mathematical Practice Cents and the Central Limit Theorem Overview of Lesson In this lesson, students conduct a hands-on demonstration of the Central Limit Theorem. They construct a distribution of a population and then construct

More information

Standard Deviation Estimator

Standard Deviation Estimator CSS.com Chapter 905 Standard Deviation Estimator Introduction Even though it is not of primary interest, an estimate of the standard deviation (SD) is needed when calculating the power or sample size of

More information

Chapter 7 Part 2. Hypothesis testing Power

Chapter 7 Part 2. Hypothesis testing Power Chapter 7 Part 2 Hypothesis testing Power November 6, 2008 All of the normal curves in this handout are sampling distributions Goal: To understand the process of hypothesis testing and the relationship

More information

Simple Regression Theory II 2010 Samuel L. Baker

Simple Regression Theory II 2010 Samuel L. Baker SIMPLE REGRESSION THEORY II 1 Simple Regression Theory II 2010 Samuel L. Baker Assessing how good the regression equation is likely to be Assignment 1A gets into drawing inferences about how close the

More information

Def: The standard normal distribution is a normal probability distribution that has a mean of 0 and a standard deviation of 1.

Def: The standard normal distribution is a normal probability distribution that has a mean of 0 and a standard deviation of 1. Lecture 6: Chapter 6: Normal Probability Distributions A normal distribution is a continuous probability distribution for a random variable x. The graph of a normal distribution is called the normal curve.

More information

GrowingKnowing.com 2011

GrowingKnowing.com 2011 GrowingKnowing.com 2011 GrowingKnowing.com 2011 1 Estimates We are often asked to predict the future! When will you complete your team project? When will you make your first million dollars? When will

More information

Inferential Statistics

Inferential Statistics Inferential Statistics Sampling and the normal distribution Z-scores Confidence levels and intervals Hypothesis testing Commonly used statistical methods Inferential Statistics Descriptive statistics are

More information

A Summary of Error Propagation

A Summary of Error Propagation A Summary of Error Propagation Suppose you measure some quantities a, b, c,... with uncertainties δa, δb, δc,.... Now you want to calculate some other quantity Q which depends on a and b and so forth.

More information

Estimation and Confidence Intervals

Estimation and Confidence Intervals Estimation and Confidence Intervals Fall 2001 Professor Paul Glasserman B6014: Managerial Statistics 403 Uris Hall Properties of Point Estimates 1 We have already encountered two point estimators: th e

More information

Review the following from Chapter 5

Review the following from Chapter 5 Bluman, Chapter 6 1 Review the following from Chapter 5 A surgical procedure has an 85% chance of success and a doctor performs the procedure on 10 patients, find the following: a) The probability that

More information

32 Measures of Central Tendency and Dispersion

32 Measures of Central Tendency and Dispersion 32 Measures of Central Tendency and Dispersion In this section we discuss two important aspects of data which are its center and its spread. The mean, median, and the mode are measures of central tendency

More information

When σ Is Known: Recall the Mystery Mean Activity where x bar = 240.79 and we have an SRS of size 16

When σ Is Known: Recall the Mystery Mean Activity where x bar = 240.79 and we have an SRS of size 16 8.3 ESTIMATING A POPULATION MEAN When σ Is Known: Recall the Mystery Mean Activity where x bar = 240.79 and we have an SRS of size 16 Task was to estimate the mean when we know that the situation is Normal

More information

Two-sample inference: Continuous data

Two-sample inference: Continuous data Two-sample inference: Continuous data Patrick Breheny April 5 Patrick Breheny STA 580: Biostatistics I 1/32 Introduction Our next two lectures will deal with two-sample inference for continuous data As

More information

4. Continuous Random Variables, the Pareto and Normal Distributions

4. Continuous Random Variables, the Pareto and Normal Distributions 4. Continuous Random Variables, the Pareto and Normal Distributions A continuous random variable X can take any value in a given range (e.g. height, weight, age). The distribution of a continuous random

More information

6 3 The Standard Normal Distribution

6 3 The Standard Normal Distribution 290 Chapter 6 The Normal Distribution Figure 6 5 Areas Under a Normal Distribution Curve 34.13% 34.13% 2.28% 13.59% 13.59% 2.28% 3 2 1 + 1 + 2 + 3 About 68% About 95% About 99.7% 6 3 The Distribution Since

More information

Chapter 4 Online Appendix: The Mathematics of Utility Functions

Chapter 4 Online Appendix: The Mathematics of Utility Functions Chapter 4 Online Appendix: The Mathematics of Utility Functions We saw in the text that utility functions and indifference curves are different ways to represent a consumer s preferences. Calculus can

More information

Sampling Distribution of a Normal Variable

Sampling Distribution of a Normal Variable Ismor Fischer, 5/9/01 5.-1 5. Formal Statement and Examples Comments: Sampling Distribution of a Normal Variable Given a random variable. Suppose that the population distribution of is known to be normal,

More information

Normal distribution. ) 2 /2σ. 2π σ

Normal distribution. ) 2 /2σ. 2π σ Normal distribution The normal distribution is the most widely known and used of all distributions. Because the normal distribution approximates many natural phenomena so well, it has developed into a

More information

Outline. Definitions Descriptive vs. Inferential Statistics The t-test - One-sample t-test

Outline. Definitions Descriptive vs. Inferential Statistics The t-test - One-sample t-test The t-test Outline Definitions Descriptive vs. Inferential Statistics The t-test - One-sample t-test - Dependent (related) groups t-test - Independent (unrelated) groups t-test Comparing means Correlation

More information

Frequency Distributions

Frequency Distributions Descriptive Statistics Dr. Tom Pierce Department of Psychology Radford University Descriptive statistics comprise a collection of techniques for better understanding what the people in a group look like

More information

z-scores AND THE NORMAL CURVE MODEL

z-scores AND THE NORMAL CURVE MODEL z-scores AND THE NORMAL CURVE MODEL 1 Understanding z-scores 2 z-scores A z-score is a location on the distribution. A z- score also automatically communicates the raw score s distance from the mean A

More information

2 ESTIMATION. Objectives. 2.0 Introduction

2 ESTIMATION. Objectives. 2.0 Introduction 2 ESTIMATION Chapter 2 Estimation Objectives After studying this chapter you should be able to calculate confidence intervals for the mean of a normal distribution with unknown variance; be able to calculate

More information

5.1 Identifying the Target Parameter

5.1 Identifying the Target Parameter University of California, Davis Department of Statistics Summer Session II Statistics 13 August 20, 2012 Date of latest update: August 20 Lecture 5: Estimation with Confidence intervals 5.1 Identifying

More information

Simple Regression Theory I 2010 Samuel L. Baker

Simple Regression Theory I 2010 Samuel L. Baker SIMPLE REGRESSION THEORY I 1 Simple Regression Theory I 2010 Samuel L. Baker Regression analysis lets you use data to explain and predict. A simple regression line drawn through data points In Assignment

More information

Week 4: Standard Error and Confidence Intervals

Week 4: Standard Error and Confidence Intervals Health Sciences M.Sc. Programme Applied Biostatistics Week 4: Standard Error and Confidence Intervals Sampling Most research data come from subjects we think of as samples drawn from a larger population.

More information

THE FIRST SET OF EXAMPLES USE SUMMARY DATA... EXAMPLE 7.2, PAGE 227 DESCRIBES A PROBLEM AND A HYPOTHESIS TEST IS PERFORMED IN EXAMPLE 7.

THE FIRST SET OF EXAMPLES USE SUMMARY DATA... EXAMPLE 7.2, PAGE 227 DESCRIBES A PROBLEM AND A HYPOTHESIS TEST IS PERFORMED IN EXAMPLE 7. THERE ARE TWO WAYS TO DO HYPOTHESIS TESTING WITH STATCRUNCH: WITH SUMMARY DATA (AS IN EXAMPLE 7.17, PAGE 236, IN ROSNER); WITH THE ORIGINAL DATA (AS IN EXAMPLE 8.5, PAGE 301 IN ROSNER THAT USES DATA FROM

More information

Z-table p-values: use choice 2: normalcdf(

Z-table p-values: use choice 2: normalcdf( P-values with the Ti83/Ti84 Note: The majority of the commands used in this handout can be found under the DISTR menu which you can access by pressing [ nd ] [VARS]. You should see the following: NOTE:

More information

Week 3&4: Z tables and the Sampling Distribution of X

Week 3&4: Z tables and the Sampling Distribution of X Week 3&4: Z tables and the Sampling Distribution of X 2 / 36 The Standard Normal Distribution, or Z Distribution, is the distribution of a random variable, Z N(0, 1 2 ). The distribution of any other normal

More information

Module 5 Hypotheses Tests: Comparing Two Groups

Module 5 Hypotheses Tests: Comparing Two Groups Module 5 Hypotheses Tests: Comparing Two Groups Objective: In medical research, we often compare the outcomes between two groups of patients, namely exposed and unexposed groups. At the completion of this

More information

ECON 600 Lecture 3: Profit Maximization Π = TR TC

ECON 600 Lecture 3: Profit Maximization Π = TR TC ECON 600 Lecture 3: Profit Maximization I. The Concept of Profit Maximization Profit is defined as total revenue minus total cost. Π = TR TC (We use Π to stand for profit because we use P for something

More information

Models for Discrete Variables

Models for Discrete Variables Probability Models for Discrete Variables Our study of probability begins much as any data analysis does: What is the distribution of the data? Histograms, boxplots, percentiles, means, standard deviations

More information

1. How different is the t distribution from the normal?

1. How different is the t distribution from the normal? Statistics 101 106 Lecture 7 (20 October 98) c David Pollard Page 1 Read M&M 7.1 and 7.2, ignoring starred parts. Reread M&M 3.2. The effects of estimated variances on normal approximations. t-distributions.

More information

Hypothesis Testing (unknown σ)

Hypothesis Testing (unknown σ) Hypothesis Testing (unknown σ) Business Statistics Recall: Plan for Today Null and Alternative Hypotheses Types of errors: type I, type II Types of correct decisions: type A, type B Level of Significance

More information

Unit 29 Chi-Square Goodness-of-Fit Test

Unit 29 Chi-Square Goodness-of-Fit Test Unit 29 Chi-Square Goodness-of-Fit Test Objectives: To perform the chi-square hypothesis test concerning proportions corresponding to more than two categories of a qualitative variable To perform the Bonferroni

More information

Math 2015 Lesson 21. We discuss the mean and the median, two important statistics about a distribution. p(x)dx = 0.5

Math 2015 Lesson 21. We discuss the mean and the median, two important statistics about a distribution. p(x)dx = 0.5 ean and edian We discuss the mean and the median, two important statistics about a distribution. The edian The median is the halfway point of a distribution. It is the point where half the population has

More information

Lesson 17: Margin of Error When Estimating a Population Proportion

Lesson 17: Margin of Error When Estimating a Population Proportion Margin of Error When Estimating a Population Proportion Classwork In this lesson, you will find and interpret the standard deviation of a simulated distribution for a sample proportion and use this information

More information

Point and Interval Estimates

Point and Interval Estimates Point and Interval Estimates Suppose we want to estimate a parameter, such as p or µ, based on a finite sample of data. There are two main methods: 1. Point estimate: Summarize the sample by a single number

More information

Statistical estimation using confidence intervals

Statistical estimation using confidence intervals 0894PP_ch06 15/3/02 11:02 am Page 135 6 Statistical estimation using confidence intervals In Chapter 2, the concept of the central nature and variability of data and the methods by which these two phenomena

More information

This is Descriptive Statistics, chapter 2 from the book Beginning Statistics (index.html) (v. 1.0).

This is Descriptive Statistics, chapter 2 from the book Beginning Statistics (index.html) (v. 1.0). This is Descriptive Statistics, chapter from the book Beginning Statistics (index.html) (v..). This book is licensed under a Creative Commons by-nc-sa. (http://creativecommons.org/licenses/by-nc-sa/./)

More information

Confidence level. Most common choices are 90%, 95%, or 99%. (α = 10%), (α = 5%), (α = 1%)

Confidence level. Most common choices are 90%, 95%, or 99%. (α = 10%), (α = 5%), (α = 1%) Confidence Interval A confidence interval (or interval estimate) is a range (or an interval) of values used to estimate the true value of a population parameter. A confidence interval is sometimes abbreviated

More information

One-Way Analysis of Variance

One-Way Analysis of Variance One-Way Analysis of Variance Note: Much of the math here is tedious but straightforward. We ll skim over it in class but you should be sure to ask questions if you don t understand it. I. Overview A. We

More information

7 Hypothesis testing - one sample tests

7 Hypothesis testing - one sample tests 7 Hypothesis testing - one sample tests 7.1 Introduction Definition 7.1 A hypothesis is a statement about a population parameter. Example A hypothesis might be that the mean age of students taking MAS113X

More information

Chapter 15 Multiple Choice Questions (The answers are provided after the last question.)

Chapter 15 Multiple Choice Questions (The answers are provided after the last question.) Chapter 15 Multiple Choice Questions (The answers are provided after the last question.) 1. What is the median of the following set of scores? 18, 6, 12, 10, 14? a. 10 b. 14 c. 18 d. 12 2. Approximately

More information

Selected Nonparametric and Parametric Statistical Tests for Two-Sample Cases 1

Selected Nonparametric and Parametric Statistical Tests for Two-Sample Cases 1 Selected Nonparametric and Parametric Statistical Tests for Two-Sample Cases The T-statistic is used to test differences in the means of two groups. The grouping variable is categorical and data for the

More information

MBA 611 STATISTICS AND QUANTITATIVE METHODS

MBA 611 STATISTICS AND QUANTITATIVE METHODS MBA 611 STATISTICS AND QUANTITATIVE METHODS Part I. Review of Basic Statistics (Chapters 1-11) A. Introduction (Chapter 1) Uncertainty: Decisions are often based on incomplete information from uncertain

More information

Report of for Chapter 2 pretest

Report of for Chapter 2 pretest Report of for Chapter 2 pretest Exam: Chapter 2 pretest Category: Organizing and Graphing Data 1. "For our study of driving habits, we recorded the speed of every fifth vehicle on Drury Lane. Nearly every

More information

Chapter 14: 1-6, 9, 12; Chapter 15: 8 Solutions When is it appropriate to use the normal approximation to the binomial distribution?

Chapter 14: 1-6, 9, 12; Chapter 15: 8 Solutions When is it appropriate to use the normal approximation to the binomial distribution? Chapter 14: 1-6, 9, 1; Chapter 15: 8 Solutions 14-1 When is it appropriate to use the normal approximation to the binomial distribution? The usual recommendation is that the approximation is good if np

More information

Probability and Statistics Lecture 9: 1 and 2-Sample Estimation

Probability and Statistics Lecture 9: 1 and 2-Sample Estimation Probability and Statistics Lecture 9: 1 and -Sample Estimation to accompany Probability and Statistics for Engineers and Scientists Fatih Cavdur Introduction A statistic θ is said to be an unbiased estimator

More information

CALCULATIONS & STATISTICS

CALCULATIONS & STATISTICS CALCULATIONS & STATISTICS CALCULATION OF SCORES Conversion of 1-5 scale to 0-100 scores When you look at your report, you will notice that the scores are reported on a 0-100 scale, even though respondents

More information

Statistics I for QBIC. Contents and Objectives. Chapters 1 7. Revised: August 2013

Statistics I for QBIC. Contents and Objectives. Chapters 1 7. Revised: August 2013 Statistics I for QBIC Text Book: Biostatistics, 10 th edition, by Daniel & Cross Contents and Objectives Chapters 1 7 Revised: August 2013 Chapter 1: Nature of Statistics (sections 1.1-1.6) Objectives

More information

T O P I C 1 2 Techniques and tools for data analysis Preview Introduction In chapter 3 of Statistics In A Day different combinations of numbers and types of variables are presented. We go through these

More information

Chapter 2 Formulas and Decimals

Chapter 2 Formulas and Decimals Chapter Formulas and Decimals Section A Rounding, Comparing, Adding and Subtracting Decimals Look at the following formulas. The first formula (P = A + B + C) is one we use to calculate perimeter of a

More information

Lesson 1: Comparison of Population Means Part c: Comparison of Two- Means

Lesson 1: Comparison of Population Means Part c: Comparison of Two- Means Lesson : Comparison of Population Means Part c: Comparison of Two- Means Welcome to lesson c. This third lesson of lesson will discuss hypothesis testing for two independent means. Steps in Hypothesis

More information

Some Notes on Taylor Polynomials and Taylor Series

Some Notes on Taylor Polynomials and Taylor Series Some Notes on Taylor Polynomials and Taylor Series Mark MacLean October 3, 27 UBC s courses MATH /8 and MATH introduce students to the ideas of Taylor polynomials and Taylor series in a fairly limited

More information

TImath.com. Statistics. Areas in Intervals

TImath.com. Statistics. Areas in Intervals Areas in Intervals ID: 9472 TImath.com Time required 30 minutes Activity Overview In this activity, students use several methods to determine the probability of a given normally distributed value being

More information

Introductory Statistics Notes

Introductory Statistics Notes Introductory Statistics Notes Jamie DeCoster Department of Psychology University of Alabama 348 Gordon Palmer Hall Box 870348 Tuscaloosa, AL 35487-0348 Phone: (205) 348-4431 Fax: (205) 348-8648 August

More information

Descriptive Statistics and Measurement Scales

Descriptive Statistics and Measurement Scales Descriptive Statistics 1 Descriptive Statistics and Measurement Scales Descriptive statistics are used to describe the basic features of the data in a study. They provide simple summaries about the sample

More information

Evaluating trigonometric functions

Evaluating trigonometric functions MATH 1110 009-09-06 Evaluating trigonometric functions Remark. Throughout this document, remember the angle measurement convention, which states that if the measurement of an angle appears without units,

More information

How to Conduct a Hypothesis Test

How to Conduct a Hypothesis Test How to Conduct a Hypothesis Test The idea of hypothesis testing is relatively straightforward. In various studies we observe certain events. We must ask, is the event due to chance alone, or is there some

More information

Notes on Continuous Random Variables

Notes on Continuous Random Variables Notes on Continuous Random Variables Continuous random variables are random quantities that are measured on a continuous scale. They can usually take on any value over some interval, which distinguishes

More information

Hypothesis Testing. James H. Steiger. Department of Psychology and Human Development Vanderbilt University

Hypothesis Testing. James H. Steiger. Department of Psychology and Human Development Vanderbilt University Department of Psychology and Human Development Vanderbilt University 1 Introduction 2 3 4 5 6 7 8 Introduction In the preceding lecture, we learned that, over repeated samples, the sample mean M based

More information

Pythagorean Triples. Chapter 2. a 2 + b 2 = c 2

Pythagorean Triples. Chapter 2. a 2 + b 2 = c 2 Chapter Pythagorean Triples The Pythagorean Theorem, that beloved formula of all high school geometry students, says that the sum of the squares of the sides of a right triangle equals the square of the

More information

Chapter 3: Data Description Numerical Methods

Chapter 3: Data Description Numerical Methods Chapter 3: Data Description Numerical Methods Learning Objectives Upon successful completion of Chapter 3, you will be able to: Summarize data using measures of central tendency, such as the mean, median,

More information

WHERE DOES THE 10% CONDITION COME FROM?

WHERE DOES THE 10% CONDITION COME FROM? 1 WHERE DOES THE 10% CONDITION COME FROM? The text has mentioned The 10% Condition (at least) twice so far: p. 407 Bernoulli trials must be independent. If that assumption is violated, it is still okay

More information

Social Studies 201 Notes for November 19, 2003

Social Studies 201 Notes for November 19, 2003 1 Social Studies 201 Notes for November 19, 2003 Determining sample size for estimation of a population proportion Section 8.6.2, p. 541. As indicated in the notes for November 17, when sample size is

More information

17.0 Linear Regression

17.0 Linear Regression 17.0 Linear Regression 1 Answer Questions Lines Correlation Regression 17.1 Lines The algebraic equation for a line is Y = β 0 + β 1 X 2 The use of coordinate axes to show functional relationships was

More information

Stats for Strategy Exam 1 In-Class Practice Questions DIRECTIONS

Stats for Strategy Exam 1 In-Class Practice Questions DIRECTIONS Stats for Strategy Exam 1 In-Class Practice Questions DIRECTIONS Choose the single best answer for each question. Discuss questions with classmates, TAs and Professor Whitten. Raise your hand to check

More information

Characteristics of Binomial Distributions

Characteristics of Binomial Distributions Lesson2 Characteristics of Binomial Distributions In the last lesson, you constructed several binomial distributions, observed their shapes, and estimated their means and standard deviations. In Investigation

More information

TEACHER NOTES MATH NSPIRED

TEACHER NOTES MATH NSPIRED Math Objectives Students will understand that normal distributions can be used to approximate binomial distributions whenever both np and n(1 p) are sufficiently large. Students will understand that when

More information

AP Statistics 2002 Scoring Guidelines

AP Statistics 2002 Scoring Guidelines AP Statistics 2002 Scoring Guidelines The materials included in these files are intended for use by AP teachers for course and exam preparation in the classroom; permission for any other use must be sought

More information

Math 140 (4,5,6) Sample Exam II Fall 2011

Math 140 (4,5,6) Sample Exam II Fall 2011 Math 140 (4,5,6) Sample Exam II Fall 2011 Provide an appropriate response. 1) In a sample of 10 randomly selected employees, it was found that their mean height was 63.4 inches. From previous studies,

More information

99.37, 99.38, 99.38, 99.39, 99.39, 99.39, 99.39, 99.40, 99.41, 99.42 cm

99.37, 99.38, 99.38, 99.39, 99.39, 99.39, 99.39, 99.40, 99.41, 99.42 cm Error Analysis and the Gaussian Distribution In experimental science theory lives or dies based on the results of experimental evidence and thus the analysis of this evidence is a critical part of the

More information

Describing Populations Statistically: The Mean, Variance, and Standard Deviation

Describing Populations Statistically: The Mean, Variance, and Standard Deviation Describing Populations Statistically: The Mean, Variance, and Standard Deviation BIOLOGICAL VARIATION One aspect of biology that holds true for almost all species is that not every individual is exactly

More information

Null Hypothesis Significance Testing Signifcance Level, Power, t-tests Spring 2014 Jeremy Orloff and Jonathan Bloom

Null Hypothesis Significance Testing Signifcance Level, Power, t-tests Spring 2014 Jeremy Orloff and Jonathan Bloom Null Hypothesis Significance Testing Signifcance Level, Power, t-tests 18.05 Spring 2014 Jeremy Orloff and Jonathan Bloom Simple and composite hypotheses Simple hypothesis: the sampling distribution is

More information

, for x = 0, 1, 2, 3,... (4.1) (1 + 1/n) n = 2.71828... b x /x! = e b, x=0

, for x = 0, 1, 2, 3,... (4.1) (1 + 1/n) n = 2.71828... b x /x! = e b, x=0 Chapter 4 The Poisson Distribution 4.1 The Fish Distribution? The Poisson distribution is named after Simeon-Denis Poisson (1781 1840). In addition, poisson is French for fish. In this chapter we will

More information

Unit 8: Normal Calculations

Unit 8: Normal Calculations Unit 8: Normal Calculations Summary of Video In this video, we continue the discussion of normal curves that was begun in Unit 7. Recall that a normal curve is bell-shaped and completely characterized

More information

Joint Probability Distributions and Random Samples. Week 5, 2011 Stat 4570/5570 Material from Devore s book (Ed 8), and Cengage

Joint Probability Distributions and Random Samples. Week 5, 2011 Stat 4570/5570 Material from Devore s book (Ed 8), and Cengage 5 Joint Probability Distributions and Random Samples Week 5, 2011 Stat 4570/5570 Material from Devore s book (Ed 8), and Cengage Two Discrete Random Variables The probability mass function (pmf) of a single

More information

Power and Sample Size Determination

Power and Sample Size Determination Power and Sample Size Determination Bret Hanlon and Bret Larget Department of Statistics University of Wisconsin Madison November 3 8, 2011 Power 1 / 31 Experimental Design To this point in the semester,

More information

3.4 Statistical inference for 2 populations based on two samples

3.4 Statistical inference for 2 populations based on two samples 3.4 Statistical inference for 2 populations based on two samples Tests for a difference between two population means The first sample will be denoted as X 1, X 2,..., X m. The second sample will be denoted

More information

c 2008 Je rey A. Miron We have described the constraints that a consumer faces, i.e., discussed the budget constraint.

c 2008 Je rey A. Miron We have described the constraints that a consumer faces, i.e., discussed the budget constraint. Lecture 2b: Utility c 2008 Je rey A. Miron Outline: 1. Introduction 2. Utility: A De nition 3. Monotonic Transformations 4. Cardinal Utility 5. Constructing a Utility Function 6. Examples of Utility Functions

More information

Introduction to the Practice of Statistics Fifth Edition Moore, McCabe

Introduction to the Practice of Statistics Fifth Edition Moore, McCabe Introduction to the Practice of Statistics Fifth Edition Moore, McCabe Section 5.1 Homework Answers 5.7 In the proofreading setting if Exercise 5.3, what is the smallest number of misses m with P(X m)

More information