# Section 7.2 Confidence Intervals for Population Proportions

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Section 7.2 Confidence Intervals for Population Proportions 2012 Pearson Education, Inc. All rights reserved. 1 of 83

2 Section 7.2 Objectives Find a point estimate for the population proportion Construct a confidence interval for a population proportion Determine the minimum sample size required when estimating a population proportion 2012 Pearson Education, Inc. All rights reserved. 2 of 83

3 Point Estimate for Population p Population Proportion The probability of success in a single trial of a binomial experiment. Denoted by p Point Estimate for p The proportion of successes in a sample. Denoted by x number of successes in sample pˆ = = n sample size read as p hat 2012 Pearson Education, Inc. All rights reserved. 3 of 83

4 Point Estimate for Population p Estimate Population Parameter with Sample Statistic Proportion: p ˆp Point Estimate for q, the population proportion of failures Denoted by Read as q hat ˆ q = 1 ˆ p 2012 Pearson Education, Inc. All rights reserved. 4 of 83

5 Example: Point Estimate for p In a survey of 1000 U.S. adults, 662 said that it is acceptable to check personal while at work. Find a point estimate for the population proportion of U.S. adults who say it is acceptable to check personal while at work. (Adapted from Liberty Mutual) Solution: n = 1000 and x = 662 pˆ x = = n = = 66.2% 2012 Pearson Education, Inc. All rights reserved. 5 of 83

6 Confidence Intervals for p A c-confidence interval for a population proportion p ˆ p E < p< pˆ + E where E = Z α 2 pq ˆˆ n The probability that the confidence interval contains p is c Pearson Education, Inc. All rights reserved. 6 of 83

7 Constructing Confidence Intervals for p In Words 1. Identify the sample statistics n and x. 2. Find the point estimate ˆp. 3. Verify that the sampling distribution of pˆ can be approximated by a normal distribution. 4. Find the critical value z c that corresponds to the given level of confidence c. In Symbols npˆ pˆ = x n 5, nqˆ 5 Use the Standard Normal Table or technology Pearson Education, Inc. All rights reserved. 7 of 83

8 Constructing Confidence Intervals for p In Words 5. Find the margin of error E. 6. Find the left and right endpoints and form the confidence interval. In Symbols E = Z α 2 pq ˆˆ n Left endpoint: ˆp E Right endpoint: ˆp+ E Interval: pˆ E < p< pˆ + E 2012 Pearson Education, Inc. All rights reserved. 8 of 83

9 Example: Confidence Interval for p In a survey of 1000 U.S. adults, 662 said that it is acceptable to check personal while at work. Construct a 95% confidence interval for the population proportion of U.S. adults who say that it is acceptable to check personal while at work. Solution: Recall p ˆ = qˆ = 1 pˆ = = Pearson Education, Inc. All rights reserved. 9 of 83

10 Solution: Confidence Interval for p Verify the sampling distribution of can be approximated by the normal distribution npˆ = = 662 > 5 nqˆ = = 338 > 5 Margin of error: ˆp E pq ˆˆ (0.662) (0.338) = Zα 2 = n Pearson Education, Inc. All rights reserved. 10 of 83

11 Solution: Confidence Interval for p Confidence interval: Left Endpoint: pˆ E < p < Right Endpoint: pˆ = = E 2012 Pearson Education, Inc. All rights reserved. 11 of 83

12 Solution: Confidence Interval for p < p < Point estimate ˆp ˆp E ˆp+ E With 95% confidence, you can say that the population proportion of U.S. adults who say that it is acceptable to check personal while at work is between 63.3% and 69.1% Pearson Education, Inc. All rights reserved. 12 of 83

13 Determining Sample Size When p is known, n = Z α 2 E 2 2 pq ˆˆ Where Z α 2 is the value associated with the desired confidence level, and E is the desired margin of error. Round up to the next integer. 13

14 Sample Size for Estimating the Population Proportion When p is unknown, we use n = Z α 2 E

15 Example: Sample Size You are running a political campaign and wish to estimate, with 95% confidence, the population proportion of registered voters who will vote for your candidate. Your estimate must be accurate within 3% of the true population proportion. Find the minimum sample size needed if 1. no preliminary estimate is available. Solution: Because you do not have a preliminary estimate for use p ˆ = 0.5 and q ˆ = p, ˆ 2012 Pearson Education, Inc. All rights reserved. 15 of 83

16 Solution: Sample Size c = 0.95 z α/2 = 1.96 E = 0.03 n 2 [ ] 2 Z ˆˆ α 2 pq 1.96 (0.5)(0.5) = = 2 2 E (0. 03) Round up to the nearest whole number. With no preliminary estimate, the minimum sample size should be at least 1068 voters Pearson Education, Inc. All rights reserved. 16 of 83

17 Example: Sample Size You are running a political campaign and wish to estimate, with 95% confidence, the population proportion of registered voters who will vote for your candidate. Your estimate must be accurate within 3% of the true population proportion. Find the minimum sample size needed if 2. a preliminary estimate gives p ˆ = Solution: Use the preliminary estimate qˆ = 1 pˆ = = ˆ 0.31 p = 2012 Pearson Education, Inc. All rights reserved. 17 of 83

18 Solution: Sample Size c = 0.95 z α/2 = 1.96 E = 0.03 n Z ˆ ˆ α 2 pq = = 2 E 2 [ ] (0.31)(0.69) (0.03) Round up to the nearest whole number. With a preliminary estimate of p ˆ = 0.31, the minimum sample size should be at least 914 voters. Need a larger sample size if no preliminary estimate is available Pearson Education, Inc. All rights reserved. 18 of 83

19 Section 7.3 Confidence Intervals for the Mean (σ known) 2012 Pearson Education, Inc. All rights reserved. 19 of 83

20 Section 7.3 Objectives Find a point estimate and a margin of error Construct and interpret confidence intervals for the population mean Determine the minimum sample size required when estimating μ 2012 Pearson Education, Inc. All rights reserved. 20 of 83

21 Point Estimate for Population μ Point Estimate A single value estimate for a population parameter Most unbiased point estimate of the population mean μ is the sample mean x Estimate Population with Sample Parameter Statistic Mean: μ x 2012 Pearson Education, Inc. All rights reserved. 21 of 83

22 Example: Point Estimate for Population μ A social networking website allows its users to add friends, send messages, and update their personal profiles. The following represents a random sample of the number of friends for 40 users of the website. Find a point estimate of the population mean, µ. (Source: Facebook) Pearson Education, Inc. All rights reserved. 22 of 83

23 Solution: Point Estimate for Population μ The sample mean of the data is Σx 5232 x = = = n 40 Your point estimate for the mean number of friends for all users of the website is friends Pearson Education, Inc. All rights reserved. 23 of 83

24 Interval Estimate Interval estimate An interval, or range of values, used to estimate a population parameter. Left endpoint Point estimate x =130.8 ( ) Interval estimate Right endpoint How confident do we want to be that the interval estimate contains the population mean μ? 2012 Pearson Education, Inc. All rights reserved. 24 of 83

25 Level of Confidence Level of confidence (C-Level) The probability that the interval estimate contains the population parameter. α/2 /2 z z = 0 Critical values The remaining area in the tails is 1 c. c is the area under the C-level standard normal curve between the critical values. z c Use the Standard Normal Table to find the corresponding z-scores Pearson Education, Inc. All rights reserved. 25 of 83 z

26 Level of Confidence If the level of confidence is 90%, this means that we are 90% confident that the interval contains the population mean μ. C-Level = 0.90 α /2= 0.05 α/2= 0.05 z α/2 z = c z = 0 Zz α/2 c = The corresponding z-scores are ± z 2012 Pearson Education, Inc. All rights reserved. 26 of 83

27 Sampling error Sampling Error The difference between the point estimate and the actual population parameter value. For μ: the sampling error is the difference x μ μ is generally unknown varies from sample to sample x 2012 Pearson Education, Inc. All rights reserved. 27 of 83

28 Margin of error Margin of Error The greatest possible distance between the point estimate and the value of the parameter it is estimating for a given level of confidence, c. Denoted by E. E = z α /2 σ n When n 30, the sample standard deviation, s, can be used for σ. Sometimes called the maximum error of estimate or error tolerance Pearson Education, Inc. All rights reserved. 28 of 83

29 Example: Finding the Margin of Error Use the social networking website data and a 95% confidence level to find the margin of error for the mean number of friends for all users of the website. Assume the sample standard deviation is about Pearson Education, Inc. All rights reserved. 29 of 83

30 Solution: Finding the Margin of Error First find the critical values z α/2 = z 1.96 c z = 0 z α/2 = % of the area under the standard normal curve falls within 1.96 standard deviations of the mean. (You can approximate the distribution of the sample means with a normal curve by the Central Limit Theorem, because n = ) 2012 Pearson Education, Inc. All rights reserved. 30 of 83 z c z

31 Solution: Finding the Margin of Error σ E = Z Z n α 2 α 2 s n You don t know σ, but since n 30, you can use s in place of σ. You are 95% confident that the margin of error for the population mean is about 16.4 friends Pearson Education, Inc. All rights reserved. 31 of 83

32 Confidence Intervals for the Population Mean A c-confidence interval for the population mean μ x E < µ < x + E where E = z c σ n The probability that the confidence interval contains μ is c Pearson Education, Inc. All rights reserved. 32 of 83

33 Constructing Confidence Intervals for μ Finding a Confidence Interval for a Population Mean (n 30 or σ known with a normally distributed population) In Words 1. Find the sample statistics n and. x In Symbols Σx x = n 2. Specify σ, if known. Otherwise, if n 30, find the sample standard deviation s and use it as an estimate for σ. s = Σ( x x) n Pearson Education, Inc. All rights reserved. 33 of 83

34 Constructing Confidence Intervals for μ In Words 3. Find the critical value z c that corresponds to the given level of confidence. 4. Find the margin of error E. 5. Find the left and right endpoints and form the confidence interval. In Symbols Use the Standard Normal Table or technology. σ E = Zα 2 n Left endpoint: x E Right endpoint: x + E Interval: x E < µ < x + E 2012 Pearson Education, Inc. All rights reserved. 34 of 83

35 Example: Constructing a Confidence Interval Construct a 95% confidence interval for the mean number of friends for all users of the website. Solution: Recall x =130.8 and E 16.4 Left Endpoint: x E = x < μ < Right Endpoint: E = Pearson Education, Inc. All rights reserved. 35 of 83

36 Solution: Constructing a Confidence Interval < μ < With 95% confidence, you can say that the population mean number of friends is between and Pearson Education, Inc. All rights reserved. 36 of 83

37 Example: Constructing a Confidence Interval σ Known A college admissions director wishes to estimate the mean age of all students currently enrolled. In a random sample of 20 students, the mean age is found to be 22.9 years. From past studies, the standard deviation is known to be 1.5 years, and the population is normally distributed. Construct a 90% confidence interval of the population mean age Pearson Education, Inc. All rights reserved. 37 of 83

38 Solution: Constructing a Confidence Interval σ Known First find the critical values α = 0.90 α/2 = 0.05 α/2 = 0.05 z c = z c z = 0 z c = z c z z c = Pearson Education, Inc. All rights reserved. 38 of 83

39 Solution: Constructing a Confidence Interval σ Known Margin of error: σ E 1.5 = Zα 2 = n 20 Confidence interval: Left Endpoint: x = E < μ < 23.5 Right Endpoint: x + E = Pearson Education, Inc. All rights reserved. 39 of 83

40 Solution: Constructing a Confidence Interval σ Known 22.3 < μ < 23.5 Point estimate x ( ) E x + x E With 90% confidence, you can say that the mean age of all the students is between 22.3 and 23.5 years Pearson Education, Inc. All rights reserved. 40 of 83

41 Interpreting the Results μ is a fixed number. It is either in the confidence interval or not. Incorrect: There is a 90% probability that the actual mean is in the interval (22.3, 23.5). Correct: If a large number of samples is collected and a confidence interval is created for each sample, approximately 90% of these intervals will contain μ Pearson Education, Inc. All rights reserved. 41 of 83

42 Interpreting the Results The horizontal segments represent 90% confidence intervals for different samples of the same size. In the long run, 9 of every 10 such intervals will contain μ. μ 2012 Pearson Education, Inc. All rights reserved. 42 of 83

43 Sample Size Given a c-confidence level and a margin of error E, the minimum sample size n needed to estimate the population mean µ is n z c σ = E If σ is unknown, you can estimate it using s, provided you have a preliminary sample with at least 30 members Pearson Education, Inc. All rights reserved. 43 of 83

44 Example: Sample Size You want to estimate the mean number of friends for all users of the website. How many users must be included in the sample if you want to be 95% confident that the sample mean is within seven friends of the population mean? Assume the sample standard deviation is about Pearson Education, Inc. All rights reserved. 44 of 83

45 Solution: Sample Size First find the critical values z c = z 1.96 c z = 0 z c = 1.96 z c z z c = Pearson Education, Inc. All rights reserved. 45 of 83

46 Solution: Sample Size z c = 1.96 σ s 53.0 E = 7 n z c σ = E When necessary, round up to obtain a whole number. You should include at least 221 users in your sample Pearson Education, Inc. All rights reserved. 46 of 83

47 Section 7.3 Summary Found a point estimate and a margin of error Constructed and interpreted confidence intervals for the population mean Determined the minimum sample size required when estimating μ 2012 Pearson Education, Inc. All rights reserved. 47 of 83

48 Section 7.4 Confidence Intervals for the Mean (σ unknown) 2012 Pearson Education, Inc. All rights reserved. 48 of 83

49 Section 7.4 Objectives Interpret the t-distribution and use a t-distribution table Construct confidence intervals when n < 30, the population is normally distributed, and σ is unknown 2012 Pearson Education, Inc. All rights reserved. 49 of 83

50 The t-distribution When the population standard deviation is unknown, the sample size is less than 30, and the random variable x is approximately normally distributed, it follows a t-distribution. t = x µ s n Critical values of t are denoted by t c Pearson Education, Inc. All rights reserved. 50 of 83

51 Properties of the t-distribution 1. The t-distribution is bell shaped and symmetric about the mean. 2. The t-distribution is a family of curves, each determined by a parameter called the degrees of freedom. The degrees of freedom are the number of free choices left after a sample statistic such as x is calculated. When you use a t-distribution to estimate a population mean, the degrees of freedom are equal to one less than the sample size. d.f. = n 1 Degrees of freedom 2012 Pearson Education, Inc. All rights reserved. 51 of 83

52 Properties of the t-distribution 3. The total area under a t-curve is 1 or 100%. 4. The mean, median, and mode of the t-distribution are equal to zero. 5. As the degrees of freedom increase, the t-distribution approaches the normal distribution. After 30 d.f., the t- distribution is very close to the standard normal z- distribution. d.f. = 2 d.f. = 5 Standard normal curve 0 t The tails in the t- distribution are thicker than those in the standard normal distribution Pearson Education, Inc. All rights reserved. 52 of 83

53 Example: Critical Values of t Find the critical value t c for a 95% confidence level when the sample size is 15. Solution: d.f. = n 1 = 15 1 = 14 Table 5: t-distribution t c = Pearson Education, Inc. All rights reserved. 53 of 83

54 Solution: Critical Values of t 95% of the area under the t-distribution curve with 14 degrees of freedom lies between t = ± c = 0.95 t c = t c = t 2012 Pearson Education, Inc. All rights reserved. 54 of 83

55 Confidence Intervals for the Population Mean A c-confidence interval for the population mean μ s x E < µ < x + E where E = tc n The probability that the confidence interval contains μ is c Pearson Education, Inc. All rights reserved. 55 of 83

56 Confidence Intervals and t-distributions In Words 1. Identify the sample statistics n, x, and s. 2. Identify the degrees of freedom, the level of confidence c, and the critical value t c. In Symbols Σx x = s = n d.f. = n 1 ( x x) n Find the margin of error E. E = t c s n 2012 Pearson Education, Inc. All rights reserved. 56 of 83

57 Confidence Intervals and t-distributions In Words 4. Find the left and right endpoints and form the confidence interval. In Symbols Left endpoint: x Right endpoint: x + Interval: x E < µ < x + E E E 2012 Pearson Education, Inc. All rights reserved. 57 of 83

58 Example: Constructing a Confidence Interval You randomly select 16 coffee shops and measure the temperature of the coffee sold at each. The sample mean temperature is 162.0ºF with a sample standard deviation of 10.0ºF. Find the 95% confidence interval for the population mean temperature. Assume the temperatures are approximately normally distributed. Solution: Use the t-distribution (n < 30, σ is unknown, temperatures are approximately normally distributed) Pearson Education, Inc. All rights reserved. 58 of 83

59 Solution: Constructing a Confidence Interval n =16, x = s = 10.0 c = 0.95 df = n 1 = 16 1 = 15 Critical Value Table 5: t-distribution t c = Pearson Education, Inc. All rights reserved. 59 of 83

60 Solution: Constructing a Confidence Interval Margin of error: s 10 E = t c = n 16 Confidence interval: Left Endpoint: x E = < μ < Right Endpoint: x + E = Pearson Education, Inc. All rights reserved. 60 of 83

61 Solution: Constructing a Confidence Interval < μ < Point estimate x ( ) E + x x E With 95% confidence, you can say that the population mean temperature of coffee sold is between 156.7ºF and 167.3ºF Pearson Education, Inc. All rights reserved. 61 of 83

62 Normal or t-distribution? Is n 30? No Is the population normally, or approximately normally, distributed? Yes Yes No Use the normal distribution with σ E = z c n If σ is unknown, use s instead. Cannot use the normal distribution or the t-distribution. Is σ known? No Use the t-distribution with s E = t c and n 1 degrees of freedom. n Yes Use the normal distribution with σ E = z c. n 2012 Pearson Education, Inc. All rights reserved. 62 of 83

63 Example: Normal or t-distribution? You randomly select 25 newly constructed houses. The sample mean construction cost is \$181,000 and the population standard deviation is \$28,000. Assuming construction costs are normally distributed, should you use the normal distribution, the t-distribution, or neither to construct a 95% confidence interval for the population mean construction cost? Solution: Use the normal distribution (the population is normally distributed and the population standard deviation is known) 2012 Pearson Education, Inc. All rights reserved. 63 of 83

64 Section 7.3 Summary Interpreted the t-distribution and used a t-distribution table Constructed confidence intervals when n < 30, the population is normally distributed, and σ is unknown 2012 Pearson Education, Inc. All rights reserved. 64 of 83

65 Section 7.3 Summary Found a point estimate for the population proportion Constructed a confidence interval for a population proportion Determined the minimum sample size required when estimating a population proportion 2012 Pearson Education, Inc. All rights reserved. 65 of 83

66 Section 7.5 Confidence Intervals for Variance and Standard Deviation 2012 Pearson Education, Inc. All rights reserved. 66 of 83

67 Section 7.5 Objectives Interpret the chi-square distribution and use a chi-square distribution table Use the chi-square distribution to construct a confidence interval for the variance and standard deviation 2012 Pearson Education, Inc. All rights reserved. 67 of 83

68 The Chi-Square Distribution The point estimate for σ 2 is s 2 The point estimate for σ is s s 2 is the most unbiased estimate for σ 2 Estimate Population Parameter with Sample Statistic Variance: σ 2 s 2 Standard deviation: σ s 2012 Pearson Education, Inc. All rights reserved. 68 of 83

69 The Chi-Square Distribution You can use the chi-square distribution to construct a confidence interval for the variance and standard deviation. If the random variable x has a normal distribution, then the distribution of 2 2 ( n 1) s χ = 2 σ forms a chi-square distribution for samples of any size n > Pearson Education, Inc. All rights reserved. 69 of 83

70 Properties of The Chi-Square Distribution 1. All chi-square values χ 2 are greater than or equal to zero. 2. The chi-square distribution is a family of curves, each determined by the degrees of freedom. To form a confidence interval for σ 2, use the χ 2 -distribution with degrees of freedom equal to one less than the sample size. d.f. = n 1 Degrees of freedom 3. The area under each curve of the chi-square distribution equals one Pearson Education, Inc. All rights reserved. 70 of 83

71 Properties of The Chi-Square Distribution 4. Chi-square distributions are positively skewed. Chi-square Distributions 2012 Pearson Education, Inc. All rights reserved. 71 of 83

72 Critical Values for χ 2 There are two critical values for each level of confidence. The value χ 2 R represents the right-tail critical value The value χ 2 L represents the left-tail critical value. 1 c 2 c 1 c 2 The area between the left and right critical values is c. 2 χ L 2 χ R χ Pearson Education, Inc. All rights reserved. 72 of 83

73 Example: Finding Critical Values for χ Find the critical values χ R and χ L for a 95% confidence interval when the sample size is 18. Solution: d.f. = n 1 = 18 1 = 17 d.f. Each area in the table represents the region under the chi-square curve to the right of the critical value. Area to the right of χ 2 R = Area to the right of χ 2 L = 1 c = = c = = Pearson Education, Inc. All rights reserved. 73 of 83

74 Solution: Finding Critical Values for χ 2 Table 6: χ 2 -Distribution 2 χ L = χ R = % of the area under the curve lies between and Pearson Education, Inc. All rights reserved. 74 of 83

75 Confidence Intervals for σ 2 and σ Confidence Interval for σ 2 : ( n 1) s ( n 1) s χ < σ < 2 R χl Confidence Interval for σ: 2 2 ( n 1) s ( n 1) s < σ < χ 2 2 R χl The probability that the confidence intervals contain σ 2 or σ is c Pearson Education, Inc. All rights reserved. 75 of 83

76 Confidence Intervals for σ 2 and σ In Words 1. Verify that the population has a normal distribution. 2. Identify the sample statistic n and the degrees of freedom. 3. Find the point estimate s Find the critical values χ 2 R and χ 2 L that correspond to the given level of confidence c. In Symbols s d.f. = n ( x x) = n 1 Use Table 6 in Appendix B Pearson Education, Inc. All rights reserved. 76 of 83

77 Confidence Intervals for σ 2 and σ In Words 5. Find the left and right endpoints and form the confidence interval for the population variance. 6. Find the confidence interval for the population standard deviation by taking the square root of each endpoint. In Symbols < σ < 2 R χl ( n 1) s ( n 1) s χ 2 2 ( n 1) s ( n 1) s < σ < χ 2 2 R χl 2012 Pearson Education, Inc. All rights reserved. 77 of 83

78 Example: Constructing a Confidence Interval You randomly select and weigh 30 samples of an allergy medicine. The sample standard deviation is 1.20 milligrams. Assuming the weights are normally distributed, construct 99% confidence intervals for the population variance and standard deviation. Solution: d.f. = n 1 = 30 1 = 29 d.f Pearson Education, Inc. All rights reserved. 78 of 83

79 Solution: Constructing a Confidence Interval Area to the right of χ 2 R = 1 c = = Area to the right of χ 2 L = 1+ c = = The critical values are χ 2 R = and χ 2 L = Pearson Education, Inc. All rights reserved. 79 of 83

80 Solution: Constructing a Confidence Interval Confidence Interval for σ 2 : Left endpoint: ( n 1) s χ 2 R 2 = (30 1)(1.20) Right endpoint: ( n 1) s χ 2 L 2 = (30 1)(1.20) < σ 2 < 3.18 With 99% confidence, you can say that the population variance is between 0.80 and Pearson Education, Inc. All rights reserved. 80 of 83

81 Solution: Constructing a Confidence Interval Confidence Interval for σ : 2 2 ( n 1) s ( n 1) s < σ < χ 2 2 R χl 2 2 (30 1)(1.20) (30 1)(1.20) < σ < < σ < 1.78 With 99% confidence, you can say that the population standard deviation is between 0.89 and 1.78 milligrams Pearson Education, Inc. All rights reserved. 81 of 83

82 Section 7.5 Summary Interpreted the chi-square distribution and used a chi-square distribution table Used the chi-square distribution to construct a confidence interval for the variance and standard deviation 2012 Pearson Education, Inc. All rights reserved. 82 of 83

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

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.

### Chapter Additional: Standard Deviation and Chi- Square

Chapter Additional: Standard Deviation and Chi- Square Chapter Outline: 6.4 Confidence Intervals for the Standard Deviation 7.5 Hypothesis testing for Standard Deviation Section 6.4 Objectives Interpret

### 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

### 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

### 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

### 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

### 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,

### 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

### 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

### 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

### 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

### 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

### MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

STATISTICS/GRACEY EXAM 3 PRACTICE/CH. 8-9 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the P-value for the indicated hypothesis test. 1) A

### Chapter 7. Section Introduction to Hypothesis Testing

Section 7.1 - Introduction to Hypothesis Testing Chapter 7 Objectives: State a null hypothesis and an alternative hypothesis Identify type I and type II errors and interpret the level of significance Determine

### MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question

Stats: Test Review Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question Provide an appropriate response. ) Given H0: p 0% and Ha: p < 0%, determine

### 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

### 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

### Need for Sampling. Very large populations Destructive testing Continuous production process

Chapter 4 Sampling and Estimation Need for Sampling Very large populations Destructive testing Continuous production process The objective of sampling is to draw a valid inference about a population. 4-

### FINAL EXAM REVIEW - Fa 13

FINAL EXAM REVIEW - Fa 13 Determine which of the four levels of measurement (nominal, ordinal, interval, ratio) is most appropriate. 1) The temperatures of eight different plastic spheres. 2) The sample

### 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

### 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.

### 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

### 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

### Sample Exam #1 Elementary Statistics

Sample Exam #1 Elementary Statistics Instructions. No books, notes, or calculators are allowed. 1. Some variables that were recorded while studying diets of sharks are given below. Which of the variables

### ESTIMATION - CHAPTER 6 1

ESTIMATION - CHAPTER 6 1 PREVIOUSLY use known probability distributions (binomial, Poisson, normal) and know population parameters (mean, variance) to answer questions such as......given 20 births and

### Lecture 8. Confidence intervals and the central limit theorem

Lecture 8. Confidence intervals and the central limit theorem Mathematical Statistics and Discrete Mathematics November 25th, 2015 1 / 15 Central limit theorem Let X 1, X 2,... X n be a random sample of

### 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

### 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

### HypoTesting. Name: Class: Date: Multiple Choice Identify the choice that best completes the statement or answers the question.

Name: Class: Date: HypoTesting Multiple Choice Identify the choice that best completes the statement or answers the question. 1. A Type II error is committed if we make: a. a correct decision when the

### 6.4 Normal Distribution

Contents 6.4 Normal Distribution....................... 381 6.4.1 Characteristics of the Normal Distribution....... 381 6.4.2 The Standardized Normal Distribution......... 385 6.4.3 Meaning of Areas under

### 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.

### 5/31/2013. 6.1 Normal Distributions. Normal Distributions. Chapter 6. Distribution. The Normal Distribution. Outline. Objectives.

The Normal Distribution C H 6A P T E R The Normal Distribution Outline 6 1 6 2 Applications of the Normal Distribution 6 3 The Central Limit Theorem 6 4 The Normal Approximation to the Binomial Distribution

### Math 251, Review Questions for Test 3 Rough Answers

Math 251, Review Questions for Test 3 Rough Answers 1. (Review of some terminology from Section 7.1) In a state with 459,341 voters, a poll of 2300 voters finds that 45 percent support the Republican candidate,

### Statistics 100 Binomial and Normal Random Variables

Statistics 100 Binomial and Normal Random Variables Three different random variables with common characteristics: 1. Flip a fair coin 10 times. Let X = number of heads out of 10 flips. 2. Poll a random

### Practice Exam. 1. What is the median of this data? A) 64 B) 63.5 C) 67.5 D) 59 E) 35

Practice Exam Use the following to answer questions 1-2: A census is done in a given region. Following are the populations of the towns in that particular region (in thousands): 35, 46, 52, 63, 64, 71,

### UCLA STAT 13 Statistical Methods - Final Exam Review Solutions Chapter 7 Sampling Distributions of Estimates

UCLA STAT 13 Statistical Methods - Final Exam Review Solutions Chapter 7 Sampling Distributions of Estimates 1. (a) (i) µ µ (ii) σ σ n is exactly Normally distributed. (c) (i) is approximately Normally

### Chapter 3 Descriptive Statistics: Numerical Measures. Learning objectives

Chapter 3 Descriptive Statistics: Numerical Measures Slide 1 Learning objectives 1. Single variable Part I (Basic) 1.1. How to calculate and use the measures of location 1.. How to calculate and use the

### 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,

### 13.2 Measures of Central Tendency

13.2 Measures of Central Tendency Measures of Central Tendency For a given set of numbers, it may be desirable to have a single number to serve as a kind of representative value around which all the numbers

### Sampling and Hypothesis Testing

Population and sample Sampling and Hypothesis Testing Allin Cottrell Population : an entire set of objects or units of observation of one sort or another. Sample : subset of a population. Parameter versus

### Chapter 8 Hypothesis Testing Chapter 8 Hypothesis Testing 8-1 Overview 8-2 Basics of Hypothesis Testing

Chapter 8 Hypothesis Testing 1 Chapter 8 Hypothesis Testing 8-1 Overview 8-2 Basics of Hypothesis Testing 8-3 Testing a Claim About a Proportion 8-5 Testing a Claim About a Mean: s Not Known 8-6 Testing

### 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

### Hypothesis Testing Population Mean

Z-test About One Mean ypothesis Testing Population Mean The Z-test about a mean of population we are using is applied in the following three cases: a. The population distribution is normal and the population

### Statistics - Written Examination MEC Students - BOVISA

Statistics - Written Examination MEC Students - BOVISA Prof.ssa A. Guglielmi 26.0.2 All rights reserved. Legal action will be taken against infringement. Reproduction is prohibited without prior consent.

### A POPULATION MEAN, CONFIDENCE INTERVALS AND HYPOTHESIS TESTING

CHAPTER 5. A POPULATION MEAN, CONFIDENCE INTERVALS AND HYPOTHESIS TESTING 5.1 Concepts When a number of animals or plots are exposed to a certain treatment, we usually estimate the effect of the treatment

### Statistics and research

Statistics and research Usaneya Perngparn Chitlada Areesantichai Drug Dependence Research Center (WHOCC for Research and Training in Drug Dependence) College of Public Health Sciences Chulolongkorn University,

### EMPIRICAL FREQUENCY DISTRIBUTION

INTRODUCTION TO MEDICAL STATISTICS: Mirjana Kujundžić Tiljak EMPIRICAL FREQUENCY DISTRIBUTION observed data DISTRIBUTION - described by mathematical models 2 1 when some empirical distribution approximates

### 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

### Review. March 21, 2011. 155S7.1 2_3 Estimating a Population Proportion. Chapter 7 Estimates and Sample Sizes. Test 2 (Chapters 4, 5, & 6) Results

MAT 155 Statistical Analysis Dr. Claude Moore Cape Fear Community College Chapter 7 Estimates and Sample Sizes 7 1 Review and Preview 7 2 Estimating a Population Proportion 7 3 Estimating a Population

### 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

### 1) What is the probability that the random variable has a value greater than 2? A) 0.750 B) 0.625 C) 0.875 D) 0.700

Practice for Chapter 6 & 7 Math 227 This is merely an aid to help you study. The actual exam is not multiple choice nor is it limited to these types of questions. Using the following uniform density curve,

### Unit 26 Estimation with Confidence Intervals

Unit 26 Estimation with Confidence Intervals Objectives: To see how confidence intervals are used to estimate a population proportion, a population mean, a difference in population proportions, or a difference

### Section 7.1. Introduction to Hypothesis Testing. Schrodinger s cat quantum mechanics thought experiment (1935)

Section 7.1 Introduction to Hypothesis Testing Schrodinger s cat quantum mechanics thought experiment (1935) Statistical Hypotheses A statistical hypothesis is a claim about a population. Null hypothesis

### AP Statistics 2011 Scoring Guidelines

AP Statistics 2011 Scoring Guidelines The College Board The College Board is a not-for-profit membership association whose mission is to connect students to college success and opportunity. Founded in

### Construct a scatterplot for the given data. 2) x Answer:

Review for Test 5 STA 2023 spr 2014 Name Given the linear correlation coefficient r and the sample size n, determine the critical values of r and use your finding to state whether or not the given r represents

### 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

### 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

### Chapter 4. Probability and Probability Distributions

Chapter 4. robability and robability Distributions Importance of Knowing robability To know whether a sample is not identical to the population from which it was selected, it is necessary to assess the

### Descriptive Statistics

Descriptive Statistics Primer Descriptive statistics Central tendency Variation Relative position Relationships Calculating descriptive statistics Descriptive Statistics Purpose to describe or summarize

### 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

### Summary of Formulas and Concepts. Descriptive Statistics (Ch. 1-4)

Summary of Formulas and Concepts Descriptive Statistics (Ch. 1-4) Definitions Population: The complete set of numerical information on a particular quantity in which an investigator is interested. We assume

### Statistics 100 Sample Final Questions (Note: These are mostly multiple choice, for extra practice. Your Final Exam will NOT have any multiple choice!

Statistics 100 Sample Final Questions (Note: These are mostly multiple choice, for extra practice. Your Final Exam will NOT have any multiple choice!) Part A - Multiple Choice Indicate the best choice

### Common probability distributionsi Math 217/218 Probability and Statistics Prof. D. Joyce, 2016

Introduction. ommon probability distributionsi Math 7/8 Probability and Statistics Prof. D. Joyce, 06 I summarize here some of the more common distributions used in probability and statistics. Some are

### 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

### Null Hypothesis H 0. The null hypothesis (denoted by H 0

Hypothesis test In statistics, a hypothesis is a claim or statement about a property of a population. A hypothesis test (or test of significance) is a standard procedure for testing a claim about a property

### 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

### 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

### 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

### 6.1. Construct and Interpret Binomial Distributions. p Study probability distributions. Goal VOCABULARY. Your Notes.

6.1 Georgia Performance Standard(s) MM3D1 Your Notes Construct and Interpret Binomial Distributions Goal p Study probability distributions. VOCABULARY Random variable Discrete random variable Continuous

### Calculate and interpret confidence intervals for one population average and one population proportion.

Chapter 8 Confidence Intervals 8.1 Confidence Intervals 1 8.1.1 Student Learning Objectives By the end of this chapter, the student should be able to: Calculate and interpret confidence intervals for one

### Sample Proportions. Example 1

Sample Proportions Start with a population of units and a variable that is categorical with two categories on the basis of some attribute that each unit either succeeds in having, or fails to have. These

### Statistiek (WISB361)

Statistiek (WISB361) Final exam June 29, 2015 Schrijf uw naam op elk in te leveren vel. Schrijf ook uw studentnummer op blad 1. The maximum number of points is 100. Points distribution: 23 20 20 20 17

### AP Statistics 2010 Scoring Guidelines

AP Statistics 2010 Scoring Guidelines The College Board The College Board is a not-for-profit membership association whose mission is to connect students to college success and opportunity. Founded in

### 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,

### Chapter 9. Section Correlation

Chapter 9 Section 9.1 - Correlation Objectives: Introduce linear correlation, independent and dependent variables, and the types of correlation Find a correlation coefficient Test a population correlation

### ELEMENTARY STATISTICS

ELEMENTARY STATISTICS Study Guide Dr. Shinemin Lin Table of Contents 1. Introduction to Statistics. Descriptive Statistics 3. Probabilities and Standard Normal Distribution 4. Estimates and Sample Sizes

### HYPOTHESIS TESTING (ONE SAMPLE) - CHAPTER 7 1. used confidence intervals to answer questions such as...

HYPOTHESIS TESTING (ONE SAMPLE) - CHAPTER 7 1 PREVIOUSLY used confidence intervals to answer questions such as... You know that 0.25% of women have red/green color blindness. You conduct a study of men

### Mathematics. Probability and Statistics Curriculum Guide. Revised 2010

Mathematics Probability and Statistics Curriculum Guide Revised 2010 This page is intentionally left blank. Introduction The Mathematics Curriculum Guide serves as a guide for teachers when planning instruction

### F. Farrokhyar, MPhil, PhD, PDoc

Learning objectives Descriptive Statistics F. Farrokhyar, MPhil, PhD, PDoc To recognize different types of variables To learn how to appropriately explore your data How to display data using graphs How

### DESCRIPTIVE STATISTICS. The purpose of statistics is to condense raw data to make it easier to answer specific questions; test hypotheses.

DESCRIPTIVE STATISTICS The purpose of statistics is to condense raw data to make it easier to answer specific questions; test hypotheses. DESCRIPTIVE VS. INFERENTIAL STATISTICS Descriptive To organize,

### MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Exam Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the mean for the given sample data. 1) Frank's Furniture employees earned the following

### 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

### Hypothesis Testing I

ypothesis Testing I The testing process:. Assumption about population(s) parameter(s) is made, called null hypothesis, denoted. 2. Then the alternative is chosen (often just a negation of the null hypothesis),

### Examination 110 Probability and Statistics Examination

Examination 0 Probability and Statistics Examination Sample Examination Questions The Probability and Statistics Examination consists of 5 multiple-choice test questions. The test is a three-hour examination

### Chapter 8. Hypothesis Testing

Chapter 8 Hypothesis Testing Hypothesis In statistics, a hypothesis is a claim or statement about a property of a population. A hypothesis test (or test of significance) is a standard procedure for testing

### 1 Measures for location and dispersion of a sample

Statistical Geophysics WS 2008/09 7..2008 Christian Heumann und Helmut Küchenhoff Measures for location and dispersion of a sample Measures for location and dispersion of a sample In the following: Variable

### 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

### 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

### 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

### Answers: a. 87.5325 to 92.4675 b. 87.06 to 92.94

1. The average monthly electric bill of a random sample of 256 residents of a city is \$90 with a standard deviation of \$24. a. Construct a 90% confidence interval for the mean monthly electric bills of

### Descriptive Statistics

Chapter 2 Descriptive Statistics 2.1 Descriptive Statistics 1 2.1.1 Student Learning Objectives By the end of this chapter, the student should be able to: Display data graphically and interpret graphs:

### Key Concept. Properties

MAT 155 Statistical Analysis Dr. Claude Moore Cape Fear Community College Chapter 6 Normal Probability Distributions 6 1 Review and Preview 6 2 The Standard Normal Distribution 6 3 Applications of Normal

### Inferences About Differences Between Means Edpsy 580

Inferences About Differences Between Means Edpsy 580 Carolyn J. Anderson Department of Educational Psychology University of Illinois at Urbana-Champaign Inferences About Differences Between Means Slide

### SAMPLING & INFERENTIAL STATISTICS. Sampling is necessary to make inferences about a population.

SAMPLING & INFERENTIAL STATISTICS Sampling is necessary to make inferences about a population. SAMPLING The group that you observe or collect data from is the sample. The group that you make generalizations

### Regression Analysis: A Complete Example

Regression Analysis: A Complete Example This section works out an example that includes all the topics we have discussed so far in this chapter. A complete example of regression analysis. PhotoDisc, Inc./Getty

### Section 5 3 The Mean and Standard Deviation of a Binomial Distribution

Section 5 3 The Mean and Standard Deviation of a Binomial Distribution Previous sections required that you to find the Mean and Standard Deviation of a Binomial Distribution by using the values from a

### Study Guide for the Final Exam

Study Guide for the Final Exam When studying, remember that the computational portion of the exam will only involve new material (covered after the second midterm), that material from Exam 1 will make