Chapter 7. Estimates and Sample Size

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Chapter 7. Estimates and Sample Size"

Transcription

1 Chapter 7. Estimates and Sample Size Chapter Problem: How do we interpret a poll about global warming? Pew Research Center Poll: From what you ve read and heard, is there a solid evidence that the average temperature on earth has been increasing over the past a few decades, or not? 1501 randomly selected US adults responded 70% yes Important issues related to this poll: How can the poll be used to estimate the percentage of US adults who believe that the earth is getting warmer? How accurate is the result of 70% likely to be? Is the sample size too small? (1501/5,139,000 = % of the population) Does the method of selecting the people to be polled have much of an effect on the results? 1

2 Review Descriptive statistics 7.1 Review and Preview Two major applications of inferential statistics o Estimate the value of a population parameter o Test some claim (or hypothesis) about a population This chapter focus on Estimating important population parameters: o o o Proportion Mean Variance Methods for determining the sample sizes necessary to estimate the above parameters

3 7. Estimating a Population Proportion 1. Introduction. Why Do We Need Confidence Interval? 3. Interpreting a Confidence Interval 4. Critical Values 5. Margin of Error 6. Determining Sample Size 7. Better-Performing Confidence Intervals 3

4 7. Estimating a Population Proportion 1. Introduction Definition a point estimate is a single value (or point) used to approximate a population parameter. We will see, the sample proportion pˆ is the best point estimate of the population proportion p. Proportion, probability, and percent. We focus on proportion. We can also work with probabilities and percentages. 4

5 7. Estimating a Population Proportion 1. Introduction e.g.1 Proportion of Adults Believing in Global warming. In the chapter problem we noted that in a Pew Research Center poll, 70% of 1501 randomly selected adults in the U.S. believe in global warming, so that sample proportion is pˆ = Find the best point estimate of the proportion of all adults in the U.S. who believe in global warming. Solution. The best point estimate of p is

6 7. Estimating a Population Proportion. Why Do We Need Confidence Interval? Definition 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 as CI. Confidence interval Associated with a confidence level Confidence level is the probability 1 (or area 1 ) is the complement of confidence level E.g. for 95% confidence level, =

7 7. Estimating a Population Proportion. Why Do We Need Confidence Interval Definition The confidence level is the probability 1 that is the proportion of times that the confidence interval actually does contains the population parameter, assuming that the estimation process is repeated a large number of times. Example Here is an example of CI found later (in e.g.3), which is based on the sample data of 1501 adults polled, with 70% of them saying that they believe in global warming: The 95% confidence interval estimate of the population proportion p is < p <

8 7. Estimating a Population Proportion 3. Interpreting a Confidence Interval There are correct interpretation and incorrect creative interpretations Correct: we are 95% confident that the interval from to 0.73 actually does contain the true value of p. This means that if we were to select many different samples of size 1501 and construct the corresponding CI, 95% of them would actually contain the value of population proportion p. (success rate) Wrong: There is a 95% chance that the true value of p will fall between and

9 7. Estimating a Population Proportion 3. Interpreting a Confidence Interval At any specific point of time, the population has a fixed and constant value p, and a CI constructed from a sample either contain p or does not. A confidence level of 95% tells us that the process that we are using will, in the long run, result in confidence interval limits that contain the true population proportion 95% of time p = This confidence interval Does not contain p = Figure 7-1 CI s from 0 Different Samples 9

10 7. Estimating a Population Proportion 4. Critical Values Notation for Critical Value z and z z / z / Figure 7- Critical Value in the Standard Normal Distribution Definition a critical value is the number on the borderline separating sample statistics that are likely to occur from those that are unlikely to occur. The number z is a critical value that is a z score with the property that it separates an area of / in the right tail of the standard normal distribution. z 10

11 7. Estimating a Population Proportion 4. Critical Values e.g. Find the critical value level. z corresponding to a 95% confidence Confidence level 95% z / Figure 7-3 Finding By Table A-, we find that z z z / The total area to the left of this boundary is for a 95% CL 11

12 7. Estimating a Population Proportion 4. Critical Values We have the following brief table: Confidence Level Critical Value, z 90% % %

13 7. Estimating a Population Proportion 5. Margin of Error Definition The margin of error E, is the maximum likely (with probability 1 ) difference between the observed sample proportion pˆ and the true value of the population p. Formula 7 1 E z / pq ˆ ˆ n Margin of error for proportion 13

14 7. Estimating a Population Proportion 5. Margin of Error Requirements for this section 1) Simple random sample ) Conditions for binomial distribution are satisfied. 3) At least 5 success and at least 5 failures. (With p and q unknown, we estimate their values using sample proportion) Notation for Proportions p = population proportion pˆ x n qˆ 1 = sample proportion of x successes in a sample of size n pˆ n = number of sample values 14

15 7. Estimating a Population Proportion 5. Margin of Error Confidence Interval (or Interval Estimate) for the Population Proportion p Where The CI is often expressed in the following equivalent formats: or pˆ E pˆ p pˆ E pˆ E, pˆ E E E z / pq ˆ ˆ n Round-off Rule for CI estimate of p Round the CI limits for p to three significant digits. 15

16 7. Estimating a Population Proportion 5. Margin of Error Procedure for Constructing a Confidence Interval for p. 1) Verify that the requirements are satisfied. z / ) Find the critical value that corresponds to the desired CL. 3) Evaluate the margin or error: E z / pq ˆ ˆ n 4) Obtain the CI: pˆ E p pˆ E 5) Round the resulting CI limits to three significant digits. 16

17 7. Estimating a Population Proportion 5. Margin of Error e.g.3 Constructing a Confidence Interval: Poll Results. In the chapter problem we noted that a Pew Research poll of 1501 randomly selected U.S. adults showed that 70% of the respondents believe in global warming. n = 1501 and pˆ = a. Find the margin of error corresponding to 95% CL b. Find the 95% CI estimate of the population proportion p. c. Based on the results, can we conclude that the majority of adults believe in global warming? d. Assuming that you are a newspaper Sol. Requirements met. 17

18 7. Estimating a Population Proportion 5. Margin of Error e.g.3 (Chapter Problem, TI-84 demo). a. z 1.96, pˆ = 0.70, qˆ = 0.30, and n = 1501, so / E z pq ˆ ˆ (0.70)(0.30) / 1.96 n b = , = , so CI: < p < 0.73 or in interval notation CI = (0.677, 0.73) c. To interpret the results Based on the CI obtained in part b, it does appears that the proportion of adults who believe in global warming is greater than 50%, so we can safely conclude that the majority of adults believe in global warming. Because the limits of and 0.73 are likely to contain the true population, it appears that the proportion is a value greater than

19 7. Estimating a Population Proportion 5. Margin of Error Analyzing Polls e.g.3 Continue. When analyzing results from polls, we should consider the following: 1) The sample should be simple random sample, not an inappropriate sample (such as a voluntary response sample) ) The confidence level should be provided. (It is often 95%, but media reports often neglect to identify it) 3) The sample size should be provided. (It is usually provided by the media, but not always) 4) Except for relatively rare cases, the quality of the poll results depends on the sampling method and the size of the sample, but the size of the population is usually not a factor. 19

20 7. Estimating a Population Proportion 6. Determining Sample Size Sample Size for Estimating Proportion p Requirement: simple random sample. z When an estimate pˆ is known: Formula 7- n E z pˆ When no estimate is known: Formula 7-3 n ˆ ˆ / pq / 0. 5 E Round-off rule for Determining Sample Size If the computed sample size n is not a whole number, round it up to the whole number. 0

21 7. Estimating a Population Proportion 6. Determining Sample Size e.g.4 How many adults use the Internet? Assume that a manager for E-Bay wants to determine the current percentage of U.S. adults who now use the Internet. How many adults must be surveyed in order to be 95% confident that the sampling percentage is in error by no more than 3 percentage points? a. Use the result from a Pew Research Center poll: in 006, 73% of U.S. adults used the Internet b. Assume that we have no prior information. a. = 0.05, z / 1.96, pˆ 0.73, qˆ 0.7, E = 0.03 n = 84 samples b. n = 1068 samples 1

22 7. Estimating a Population Proportion 6. Determining Sample Size Interpretation. To be 95% confident that our sample percentage is within 3 percentage points of the true percentage for all adults, we should obtain a simple random sample of 1068 adults. By comparing this result to the sample size of 84 found in part (a), we can see that if we have no knowledge of a prior study, a larger sample is required to achieve the same result as when the value of pˆ can be estimated.

23 7. Estimating a Population Proportion 6. Determining Sample Size Finding the Point Estimate and E from a Confidence Interval. Point estimate of p: (upper confidence limit) (lower confidence limit) pˆ Margin of Error: ( upper confidence limit) (lower confidence limit) E e.g.5 Given that a confidence interval = (0.58, 0.81), Find pˆ and E pˆ E

24 7. Estimating a Population Proportion 7. Better-Performing Confidence Intervals Skip Using Technology for CI s Statdisk TI-84 (1-PropZInt, STAT/TESTS/1-PropZInt) 4

25 7.3 Estimating a Population Mean: Known 1. Introduction. Confidence Interval 3. Determining Sample Size Required to Estimate 4. Using Technology 5

26 7.3 Estimating a Population Mean: Known 1. Introduction Goal: present methods for using sample data to find a point estimate and CI estimate of population mean. Requirements: Simple random sample The population SD is known Normal distribution or n > 30. Known population standard deviation 6

27 7.3 Estimating a Population Mean: Known 1. Introduction The sample mean x is the best point estimate of the population mean. Unbiased estimator of population mean For many populations, the distribution of sample mean x tends to be more consistent (with less variation) than the distributions of other sample statistics. 7

28 7.3 Estimating a Population Mean: Known. Confidence Interval Margin of error for estimating mean ( known) is: E z / n Confidence Interval Estimate for the Population Mean (with known) is: x E x E Where E z / n or x E or ( x E, x E) 8

29 7.3 Estimating a Population Mean: Known. Confidence Interval Procedure for Constructing a Confidence Interval for p. 1) Verify that the requirements are satisfied. z / ) Find the critical value that corresponds to the desired CL. 3) Evaluate the margin or error: E z / n 4) Obtain the CI: xˆ E xˆ E 5) Rounding: next page. 9

30 7.3 Estimating a Population Mean: Known. Confidence Interval Round-Off Rule for CI Used to Estimate 1. When use the original set of data to construct a confidence interval, round the CI limits to one more decimal place than is used for the original set of data.. When use the summary statistics (n, x, s) to construct CI, round the CI limits to the same number of decimal places used for the sample mean. 30

31 7.3 Estimating a Population Mean: Known. Confidence Interval Interpreting a Confidence Interval similar to 7. for proportion be careful to interpret CI correctly. After obtaining a CI estimate of the population mean, such as 95% CI of < < Correct: we are 95% confident that the interval from to actually does contain the true value of. This means that if we were to select many different samples of the same size and construct the corresponding CI, 95% of them would actually contain the value of population mean. Wrong: There is a 95% chance that the true value of will fall between and

32 7.3 Estimating a Population Mean: Known. Confidence Interval e.g.1 Weights of Men. n = 40, x = lb, and = 6 lb. Using a 95% CL, find the following (TI-84): a. The margin of error E, and CI for. b. What do the results suggest about the mean weight of lb that was used to determine the safe passenger capacity in 1996? Sol. a. 6 E z / n 40 CI is x E x E: < < < < (round to two decimals as in x ) 3

33 7.3 Estimating a Population Mean: Known. Confidence Interval Interpretation. The confidence interval from part (a) could also be expressed in ± 8.06 or as (164.49, ). Based on the sample with n = 40 and x = and assumed to be 6, the confidence interval for the population means is lb < < lb and this interval has a 0.95 confidence level. This means that if we were to select many different simple random samples of 40 men and construct the confidence intervals as we did here, 95% of them would contain the value of the population mean 33

34 7.3 Estimating a Population Mean: Known. Confidence Interval Rationale for CI. By Central Limit Theorem, sample means (of sample size n) are normally distributed with mean and variance / n. In the equation z = ( x x ) / x, replace x with / n, replace x with, then solve for to get x z n 34

35 7.3 Estimating a Population Mean: Known 3. Determining Sample Size Required to Estimate Sample Size for Estimating mean Formula 7-4 n z E / where z / = critical z score based on the desired CL E = desired margin of error = population standard deviation Comments It does not depends on population size N. Round up. 35

36 7.3 Estimating a Population Mean: Known 3. Determining Sample Size Required to Estimate Dealing with Unknown When Finding Sample Size Use range rule of thumb: range / 4 Start the sample collection process without knowing and, using the first several values, calculate the sample standard deviation s and use it in places of. The estimated value of can then be improved as more sample data are obtained, and the sample size can be refined accordingly. Estimating the value of by using the results of some other study that was done earlier. Use other know results. E.g. IQ tests are typically designed so that the mean is 100 and SD is 15. For a population of statistics professors, IQ is more than 100 and SD is less than 15. But we can assume SD is 15 to play it safe. My comments: see next section!! 36

37 7.3 Estimating a Population Mean: Known 3. Determining Sample Size Required to Estimate e.g. IQ Scores of Statistics students. Want to estimate the mean IQ score for the population of statistics students. Q: how many statistics students must be randomly selected for IQ tests if we want 95% confidence that the sample mean is within 3 IQ points of the population mean? Sol. = 1.96 (since = 0.05) Now z / n E = 3 = 15 (see previous page) z / 1.96(15) E 3 97 samples 37

38 7.3 Estimating a Population Mean: Known 3. Determining Sample Size Required to Estimate Interpretation. Among the thousands of statistics students, we need to obtain a simple random sample of at least 97 of them. Then we get their IQ scores. With a simple random sample of only 97 statistics students, we will be 95% confident that the sample mean x is within 3 IQ points of the true population mean. 38

39 7.3 Estimating a Population Mean: Known Using Technology for CI s See e.g., TI-84 Demo Statdisk (analysis) 39

40 7.4 Estimating a Population Mean: Not Known 1. Introduction. Confidence Interval 3. Choosing the Appropriate Distribution 4. Finding Point Estimate and E from a CI 5. Using Technology 40

41 7.4 Estimating a Population Mean: Not Known 1. Introduction is not known Use Student t distribution (instead of normal distribution) Method in this section is realistic, practical, and often used Requirement Simple random sample From normally distributed population or n > 30 41

42 7.4 Estimating a Population Mean: Not Known 1. Introduction Just like in section 7.3, we have: The sample mean x is the best point estimate of the population mean. 4

43 7.4 Estimating a Population Mean: Not Known. Confidence Interval Before finding the CI, first introduce Student t distribution. Student t Distribution If a population has normal distribution, then the distribution of t x s n is a Student t Distribution for all sample of size n. A student t distribution often referred to as a t distribution, is used to find critical values denoted by. t / 43

44 7.4 Estimating a Population Mean: Not Known. Confidence Interval Definition The number of degrees of freedom for a collection of sample data is the number of sample values that can vary after certain restrictions have been imposed on all data values. For application in this section, degree of freedom = n 1 Example 1. Finding a Critical Value. n = 7 samples is selected from a normally distributed population. Find with 95% CL t / Sol. df = 7 1 = 6 Table A-3 6 th row for 95% CL, = 0.05, find the column listing values for an area of 0.05 in two tails =.447 (can do TI-84 demo). t / 44

45 7.4 Estimating a Population Mean: Not Known. Confidence Interval Margin of Error E for the Estimate of (with not known) E t / s n Confidence Interval for the Estimate (with not known) x E x E 45

46 7.4 Estimating a Population Mean: Not Known. Confidence Interval Procedure for Constructing a CI for (with not known) Step 1. Verify the requirements are met. (simple random sample, either data is from a normal distribution or n > 30) Step. Given CL, let df = n 1, use A-3 to find the critical value t / that corresponds to the desired CL Step 3. Evaluate the margin of error E t / s n Step 4. Find x. Then the CI is x E x E Step 5. Original data: add one decimal place; Summary: same. 46

47 7.4 Estimating a Population Mean: Not Known. Confidence Interval e.g. Constructing a Confidence Interval: Garlic for Reducing Cholesterol. Use summary data, n = 49, x 0.4, s = 1.0 and t /.009. The margin of error (95% CL) is: E The CI is < < , or 5.6 < < 6.4 Interpretation. Because the confidence interval contains the value of 0, it is possible that the mean of the changes in LDL cholesterol is equal to 0, suggesting that the garlic treatment did not affect the LDL cholesterol levels. It does not appear that the garlic treatment is effective in lowering LDL cholesterol. 47

48 7.4 Estimating a Population Mean: Not Known. Confidence Interval Important Properties of the Student t Distribution 1. Different for different sample size (unlike standard normal distribution). Same bell shape as the standard normal distribution. But reflect greater variability (with wider distributions) that is expected with small samples. (n = 3, wider, n = 1 narrower) 3. Mean = 0 (just like standard normal distribution) 4. SD varies with sample size. It is always greater than 1 (SD > 1). Unlike standard normal distribution where SD = As sample size gets larger, Student t distribution gets closer to standard normal distribution. 48

49 7.4 Estimating a Population Mean: Not Known 3. Choosing the Appropriate Distribution yes start Is known? no Figure 7 6 Choosing Between z and t yes Is the Population normally Distributed? no yes Is the Population normally Distributed? no yes Is n > 30? no yes Is n > 30? no Z Use the normal distribution Use nonparametric or bootstrapping methods t Use the t distribution Use nonparametric or bootstrapping methods 49

50 7.4 Estimating a Population Mean: Not Known 3. Choosing the Appropriate Distribution Method Use normal (z) distribution Use t distribution Use a nonparametric method or bootstrapping Conditions Known and normally distributed population Or Known and n > 30 not known and normally distributed population Or not known and n > 30 Population is not normally distribution and n 30 Note: 1. Criteria for deciding whether the population is normally distributed: Population need not be exactly norm, but is should appear to be somewhat symmetric with one mode no outliers. Sample size > 30: This is a commonly used guideline, but sample sizes of 15 to 30 are adequate if the population appears to have a distribution that is not far from being normal and there is no outliers. For some population distributions that are extremely far from normal, the sample size might need be much larger than

51 7.4 Estimating a Population Mean: Not Known 3. Choosing the Appropriate Distribution Example 3. Choosing Distributions. To construct CI for. Use the given data to determine whether the margin or error E should be calculated using z / (from normal distribution), t / (from Student t distribution) or neither? a. n = 9, x 75, s = 15, and population has a normal distribution t / b. n = 5, x 0, s =, and population has a very skewed distribution Neither c. n = 1, x 98.6, = 0.6, and population has a normal distribution z / d. n = 75, x 98.6, = 0.6, and distribution is skewed z / e. n = 75, x 98.6, s = 0.6, and distribution is extremely skewed t / 51

52 7.4 Estimating a Population Mean: Not Known 3. Choosing the Appropriate Distribution e.g.4 CI for Alcohol in Video Games. Twelve different video games showing substance use were observed. The duration times (in second) of alcohol use were recorded, with the times listed below. The design of the study justifies the assumption that the sample can be treated as a simple random sample. 84, 14, 583, 50, 0, 57, 07, 43, 178, 0,, 57 Use this data to construct 95% CI estimate of, the mean duration time that the video showed the use of alcohol. Sol. Next page. 5

53 Frequency 7.4 Estimating a Population Mean: Not Known 3. Choosing the Appropriate Distribution e.g.4 Continue. Check requirements: not normal, n > 30 not satisfied. Requirement are not satisfied. 5 4 (TI-84 demo, Tinterval) to get CI = (1.8, 10.7) 3 1 Interpretation. Because the requirements not satisfied, we don t have the 95% confidence that (1.8 sec, 10.7 sec) interval contains the true population mean. We should use some other method. 53

54 7.4 Estimating a Population Mean: Not Known 4. Finding Point Estimate and E from a CI Given the upper and lower limits of a CI, find the mean and margin of error: Point estimate of : x (upper confidence limit) (lower confidence limit) Margin of error: E (upper confidence limit) (lower confidence limit) e.g.5 Weights of Garbage. Data Set in Appendix B. Given 95% CI (4.8, ) Find the mean and margin of error. Sol x lb E lb 54

55 7.4 Estimating a Population Mean: Not Known 5. Using Technology TI-84 demo summary data original data Statdisk 55

56 7.5 Estimating a Population Variance 1. Chi-Square Distribution. Estimator of 56

57 7.5 Estimating a Population Variance 1. Chi-Square Distribution Given a normally distribution population with variance, randomly select a sample of size n Compute the sample variance s, The sample statistic = (n 1)s / has sampling distribution called chi-square distribution. Chi-Square Distribution Formula 7 5 Where n = sample size ( n 1) s = sample variance = population variance s 57

58 7.5 Estimating a Population Variance 1. Chi-Square Distribution Denote the chi-square by Pronounced kigh square To find the critical value, refer to A-4 Chi-square distribution depends on degree of freedom Degree of freedom df = n 1 58

59 7.5 Estimating a Population Variance 1. Chi-Square Distribution Properties of the Distribution of the Statistic Not symmetric, as df increases, it becomes more symmetric Value of can be positive or zero, but never negative The distribution is different for different df. As df increases, the distribution approaches to normal distribution (just the t distribution) 59

60 7.5 Estimating a Population Variance 1. Chi-Square Distribution e.g.1 Finding Critical Values of. A Simple random sample of ten voltage levels is obtained. Construction of a confidence level for the population standard deviation requires that the left and right critical values of corresponding to the confidence level of 95% and a sample size of n = 10. Find the critical value of separating an area of 0.05 in the left tail, and find the critical value of separating an area of 0.05 in the right tail. Sol. Df = 10 1 = 9. (next page) 60

61 7.5 Estimating a Population Variance e.g.1 continue (can also use Statdisk, Excel, and Minitab) Sol. 61

62 . Estimator of Point estimator 7.5 Estimating a Population Variance The sample variance s is the best point estimate of the population variance The sample standard deviation s is commonly used as a point estimate of (even though it is a biased estimate) 6

63 . Estimator of Interval estimator Requirements: 7.5 Estimating a Population Variance 1) The sample is a simple random sample ) The population must have normally distributed values CI for the population variance : ( n 1) s R ( n 1) s L CI for the population standard deviation: ( n 1) s R ( n 1) s L 63

64 7.5 Estimating a Population Variance. Estimator of Procedure for constructing a confidence interval for or. Step 1. verify that the requirements are satisfied Step. Using n 1 degree of freedom, Table A-4 to find critical values L and R corresponding to the desired confidence level. Step 3. Find the CI: ( n 1) s R ( n 1) s L Step 4. Find CI for. Take the square root on all three places. Step 5. Rounding. Data: add one decimal place; summary: same. 64

65 7.5 Estimating a Population Variance. Estimator of e.g. Confidence Interval for Home Voltage. Sample of 10: 13.3, 13.5, 13.7, 13.4, 13.6, 13.5, 13.5, 13.4, 13.6, 13.8 Step 1. requirements: Histogram is normal. Simple random sample. 65

66 7.5 Estimating a Population Variance. Estimator of e.g. Confidence Interval for Home Voltage. Sample of 10: 13.3, 13.5, 13.7, 13.4, 13.6, 13.5, 13.5, 13.4, 13.6, 13.8 Step. For 95% CL, double sided. Found that Step 3. R = (10 1)(0.15) or, < < Step 4. Take square root: 0.10 volt < < 0.7 volt =.700, and Interpretation: Based on this result, the confidence interval is (0.10, 0.7). The limits of the CI is 0.10 and 0.7. But the format s E cannot be used because the CI does not have s at its center. L (10 1)(0.15)

67 7.5 Estimating a Population Variance. Estimator of Rationale for the CI. If we obtain simple random samples of size n from a population with variance, there is a probability of 1 that the statistic (n 1)s / will fall between the critical values of L and R, i.e. there is a probability that the following is true: ( n 1) s Combine both in equality into one inequality, we have: ( n 1) s R R and ( n 1) s ( n L 1) s L 67

68 7.5 Estimating a Population Variance. Estimator of Determine the sample size. Harder than in the case of mean and proportion. Use Table 7- Statdisk/Analysis/Sample size determination/estimate St Dev Excel, Minitab, TI-84 does not provide sample size. 68

69 . Estimator of 7.5 Estimating a Population Variance 69

70 . Estimator of 7.5 Estimating a Population Variance e.g.3. Finding the sample size for estimating. We want to estimate the standard deviation for all voltage levels in a home. We want to be 95% confident that our estimate is within 0% of the true value of. How large should the sample size be? Assume that the population is normally distributed. Ans. at least 48 samples 70

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

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

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

Density Curve. A density curve is the graph of a continuous probability distribution. It must satisfy the following properties:

Density Curve. A density curve is the graph of a continuous probability distribution. It must satisfy the following properties: Density Curve A density curve is the graph of a continuous probability distribution. It must satisfy the following properties: 1. The total area under the curve must equal 1. 2. Every point on the curve

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

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

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

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

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

More information

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

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

Key Concept. Density Curve

Key Concept. Density Curve 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

More information

3.2 Measures of Spread

3.2 Measures of Spread 3.2 Measures of Spread In some data sets the observations are close together, while in others they are more spread out. In addition to measures of the center, it's often important to measure the spread

More information

Descriptive Statistics

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

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

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

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

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 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

More information

Biostatistics: DESCRIPTIVE STATISTICS: 2, VARIABILITY

Biostatistics: DESCRIPTIVE STATISTICS: 2, VARIABILITY Biostatistics: DESCRIPTIVE STATISTICS: 2, VARIABILITY 1. Introduction Besides arriving at an appropriate expression of an average or consensus value for observations of a population, it is important to

More information

MINITAB ASSISTANT WHITE PAPER

MINITAB ASSISTANT WHITE PAPER MINITAB ASSISTANT WHITE PAPER This paper explains the research conducted by Minitab statisticians to develop the methods and data checks used in the Assistant in Minitab 17 Statistical Software. One-Way

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

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

Chapter Additional: Standard Deviation and Chi- Square

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

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

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

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

More information

November 08, 2010. 155S8.6_3 Testing a Claim About a Standard Deviation or Variance

November 08, 2010. 155S8.6_3 Testing a Claim About a Standard Deviation or Variance Chapter 8 Hypothesis Testing 8 1 Review and Preview 8 2 Basics of Hypothesis Testing 8 3 Testing a Claim about a Proportion 8 4 Testing a Claim About a Mean: σ Known 8 5 Testing a Claim About a Mean: σ

More information

Good luck! BUSINESS STATISTICS FINAL EXAM INSTRUCTIONS. Name:

Good luck! BUSINESS STATISTICS FINAL EXAM INSTRUCTIONS. Name: Glo bal Leadership M BA BUSINESS STATISTICS FINAL EXAM Name: INSTRUCTIONS 1. Do not open this exam until instructed to do so. 2. Be sure to fill in your name before starting the exam. 3. You have two hours

More information

Sampling and Hypothesis Testing

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

More information

Estimates and Sample Sizes

Estimates and Sample Sizes 7-1 Review and Preview 7-2 Estimating a Population Proportion 7-3 Estimating a Population Mean: s Known 7-4 Estimating a Population Mean: s Not Known 7-5 Estimating a Population Variance Estimates and

More information

The Big 50 Revision Guidelines for S1

The Big 50 Revision Guidelines for S1 The Big 50 Revision Guidelines for S1 If you can understand all of these you ll do very well 1. Know what is meant by a statistical model and the Modelling cycle of continuous refinement 2. Understand

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

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

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

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,

More information

CHAPTER 11 CHI-SQUARE: NON-PARAMETRIC COMPARISONS OF FREQUENCY

CHAPTER 11 CHI-SQUARE: NON-PARAMETRIC COMPARISONS OF FREQUENCY CHAPTER 11 CHI-SQUARE: NON-PARAMETRIC COMPARISONS OF FREQUENCY The hypothesis testing statistics detailed thus far in this text have all been designed to allow comparison of the means of two or more samples

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

Lecture Notes Module 1

Lecture Notes Module 1 Lecture Notes Module 1 Study Populations A study population is a clearly defined collection of people, animals, plants, or objects. In psychological research, a study population usually consists of a specific

More information

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

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

More information

Study Guide for the Final Exam

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

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

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

Introduction to. Hypothesis Testing CHAPTER LEARNING OBJECTIVES. 1 Identify the four steps of hypothesis testing.

Introduction to. Hypothesis Testing CHAPTER LEARNING OBJECTIVES. 1 Identify the four steps of hypothesis testing. Introduction to Hypothesis Testing CHAPTER 8 LEARNING OBJECTIVES After reading this chapter, you should be able to: 1 Identify the four steps of hypothesis testing. 2 Define null hypothesis, alternative

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

6.4 Normal Distribution

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

More information

Data Mining Techniques Chapter 5: The Lure of Statistics: Data Mining Using Familiar Tools

Data Mining Techniques Chapter 5: The Lure of Statistics: Data Mining Using Familiar Tools Data Mining Techniques Chapter 5: The Lure of Statistics: Data Mining Using Familiar Tools Occam s razor.......................................................... 2 A look at data I.........................................................

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

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

Measures of Center Section 3-2 Definitions Mean (Arithmetic Mean)

Measures of Center Section 3-2 Definitions Mean (Arithmetic Mean) Measures of Center Section 3-1 Mean (Arithmetic Mean) AVERAGE the number obtained by adding the values and dividing the total by the number of values 1 Mean as a Balance Point 3 Mean as a Balance Point

More information

Means, standard deviations and. and standard errors

Means, standard deviations and. and standard errors CHAPTER 4 Means, standard deviations and standard errors 4.1 Introduction Change of units 4.2 Mean, median and mode Coefficient of variation 4.3 Measures of variation 4.4 Calculating the mean and standard

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

Statistics 2014 Scoring Guidelines

Statistics 2014 Scoring Guidelines AP Statistics 2014 Scoring Guidelines College Board, Advanced Placement Program, AP, AP Central, and the acorn logo are registered trademarks of the College Board. AP Central is the official online home

More information

FINAL EXAM REVIEW - Fa 13

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

More information

Section 12.2, Lesson 3. What Can Go Wrong in Hypothesis Testing: The Two Types of Errors and Their Probabilities

Section 12.2, Lesson 3. What Can Go Wrong in Hypothesis Testing: The Two Types of Errors and Their Probabilities Today: Section 2.2, Lesson 3: What can go wrong with hypothesis testing Section 2.4: Hypothesis tests for difference in two proportions ANNOUNCEMENTS: No discussion today. Check your grades on eee and

More information

Key Concept. Properties

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

More information

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

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

More information

TImath.com. F Distributions. Statistics

TImath.com. F Distributions. Statistics F Distributions ID: 9780 Time required 30 minutes Activity Overview In this activity, students study the characteristics of the F distribution and discuss why the distribution is not symmetric (skewed

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

2 Sample t-test (unequal sample sizes and unequal variances)

2 Sample t-test (unequal sample sizes and unequal variances) Variations of the t-test: Sample tail Sample t-test (unequal sample sizes and unequal variances) Like the last example, below we have ceramic sherd thickness measurements (in cm) of two samples representing

More information

1. What is the critical value for this 95% confidence interval? CV = z.025 = invnorm(0.025) = 1.96

1. What is the critical value for this 95% confidence interval? CV = z.025 = invnorm(0.025) = 1.96 1 Final Review 2 Review 2.1 CI 1-propZint Scenario 1 A TV manufacturer claims in its warranty brochure that in the past not more than 10 percent of its TV sets needed any repair during the first two years

More information

SPSS Workbook 4 T-tests

SPSS Workbook 4 T-tests TEESSIDE UNIVERSITY SCHOOL OF HEALTH & SOCIAL CARE SPSS Workbook 4 T-tests Research, Audit and data RMH 2023-N Module Leader:Sylvia Storey Phone:016420384969 s.storey@tees.ac.uk SPSS Workbook 4 Differences

More information

AP Statistics 2012 Scoring Guidelines

AP Statistics 2012 Scoring Guidelines AP Statistics 2012 Scoring Guidelines The College Board The College Board is a mission-driven not-for-profit organization that connects students to college success and opportunity. Founded in 1900, the

More information

Fairfield Public Schools

Fairfield Public Schools Mathematics Fairfield Public Schools AP Statistics AP Statistics BOE Approved 04/08/2014 1 AP STATISTICS Critical Areas of Focus AP Statistics is a rigorous course that offers advanced students an opportunity

More information

HYPOTHESIS TESTING: CONFIDENCE INTERVALS, T-TESTS, ANOVAS, AND REGRESSION

HYPOTHESIS TESTING: CONFIDENCE INTERVALS, T-TESTS, ANOVAS, AND REGRESSION HYPOTHESIS TESTING: CONFIDENCE INTERVALS, T-TESTS, ANOVAS, AND REGRESSION HOD 2990 10 November 2010 Lecture Background This is a lightning speed summary of introductory statistical methods for senior undergraduate

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

ELEMENTARY STATISTICS

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

More information

Chapter 8. Hypothesis Testing

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

More information

Unit 26 Estimation with Confidence Intervals

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

More information

Chi-Square Test. Contingency Tables. Contingency Tables. Chi-Square Test for Independence. Chi-Square Tests for Goodnessof-Fit

Chi-Square Test. Contingency Tables. Contingency Tables. Chi-Square Test for Independence. Chi-Square Tests for Goodnessof-Fit Chi-Square Tests 15 Chapter Chi-Square Test for Independence Chi-Square Tests for Goodness Uniform Goodness- Poisson Goodness- Goodness Test ECDF Tests (Optional) McGraw-Hill/Irwin Copyright 2009 by The

More information

AP Statistics 2011 Scoring Guidelines

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

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

Confidence Intervals for the Difference Between Two Means

Confidence Intervals for the Difference Between Two Means Chapter 47 Confidence Intervals for the Difference Between Two Means Introduction This procedure calculates the sample size necessary to achieve a specified distance from the difference in sample means

More information

Statistical Inference and t-tests

Statistical Inference and t-tests 1 Statistical Inference and t-tests Objectives Evaluate the difference between a sample mean and a target value using a one-sample t-test. Evaluate the difference between a sample mean and a target value

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

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

A POPULATION MEAN, CONFIDENCE INTERVALS AND HYPOTHESIS TESTING

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

More information

8 6 X 2 Test for a Variance or Standard Deviation

8 6 X 2 Test for a Variance or Standard Deviation Section 8 6 x 2 Test for a Variance or Standard Deviation 437 This test uses the P-value method. Therefore, it is not necessary to enter a significance level. 1. Select MegaStat>Hypothesis Tests>Proportion

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

Session 1.6 Measures of Central Tendency

Session 1.6 Measures of Central Tendency Session 1.6 Measures of Central Tendency Measures of location (Indices of central tendency) These indices locate the center of the frequency distribution curve. The mode, median, and mean are three indices

More information

Chapter 7 Section 7.1: Inference for the Mean of a Population

Chapter 7 Section 7.1: Inference for the Mean of a Population Chapter 7 Section 7.1: Inference for the Mean of a Population Now let s look at a similar situation Take an SRS of size n Normal Population : N(, ). Both and are unknown parameters. Unlike what we used

More information

F. Farrokhyar, MPhil, PhD, PDoc

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

More information

AP STATISTICS 2010 SCORING GUIDELINES

AP STATISTICS 2010 SCORING GUIDELINES 2010 SCORING GUIDELINES Question 4 Intent of Question The primary goals of this question were to (1) assess students ability to calculate an expected value and a standard deviation; (2) recognize the applicability

More information

Introduction to Statistics for Psychology. Quantitative Methods for Human Sciences

Introduction to Statistics for Psychology. Quantitative Methods for Human Sciences Introduction to Statistics for Psychology and Quantitative Methods for Human Sciences Jonathan Marchini Course Information There is website devoted to the course at http://www.stats.ox.ac.uk/ marchini/phs.html

More information

Odds ratio, Odds ratio test for independence, chi-squared statistic.

Odds ratio, Odds ratio test for independence, chi-squared statistic. Odds ratio, Odds ratio test for independence, chi-squared statistic. Announcements: Assignment 5 is live on webpage. Due Wed Aug 1 at 4:30pm. (9 days, 1 hour, 58.5 minutes ) Final exam is Aug 9. Review

More information

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

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

More information

Research Variables. Measurement. Scales of Measurement. Chapter 4: Data & the Nature of Measurement

Research Variables. Measurement. Scales of Measurement. Chapter 4: Data & the Nature of Measurement Chapter 4: Data & the Nature of Graziano, Raulin. Research Methods, a Process of Inquiry Presented by Dustin Adams Research Variables Variable Any characteristic that can take more than one form or value.

More information

1.5 Oneway Analysis of Variance

1.5 Oneway Analysis of Variance Statistics: Rosie Cornish. 200. 1.5 Oneway Analysis of Variance 1 Introduction Oneway analysis of variance (ANOVA) is used to compare several means. This method is often used in scientific or medical experiments

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

Regression Analysis: A Complete Example

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

More information

EXAM #1 (Example) Instructor: Ela Jackiewicz. Relax and good luck!

EXAM #1 (Example) Instructor: Ela Jackiewicz. Relax and good luck! STP 231 EXAM #1 (Example) Instructor: Ela Jackiewicz Honor Statement: I have neither given nor received information regarding this exam, and I will not do so until all exams have been graded and returned.

More information

SAMPLING DISTRIBUTIONS

SAMPLING DISTRIBUTIONS 0009T_c07_308-352.qd 06/03/03 20:44 Page 308 7Chapter SAMPLING DISTRIBUTIONS 7.1 Population and Sampling Distributions 7.2 Sampling and Nonsampling Errors 7.3 Mean and Standard Deviation of 7.4 Shape of

More information

STA-201-TE. 5. Measures of relationship: correlation (5%) Correlation coefficient; Pearson r; correlation and causation; proportion of common variance

STA-201-TE. 5. Measures of relationship: correlation (5%) Correlation coefficient; Pearson r; correlation and causation; proportion of common variance Principles of Statistics STA-201-TE This TECEP is an introduction to descriptive and inferential statistics. Topics include: measures of central tendency, variability, correlation, regression, hypothesis

More information

TI 83/84 Calculator The Basics of Statistical Functions

TI 83/84 Calculator The Basics of Statistical Functions What you want to do How to start What to do next Put Data in Lists STAT EDIT 1: EDIT ENTER Clear numbers already in a list: Arrow up to L1, then hit CLEAR, ENTER. Then just type the numbers into the appropriate

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

Testing Research and Statistical Hypotheses

Testing Research and Statistical Hypotheses Testing Research and Statistical Hypotheses Introduction In the last lab we analyzed metric artifact attributes such as thickness or width/thickness ratio. Those were continuous variables, which as you

More information

Using Kruskal-Wallis to Improve Customer Satisfaction. A White Paper by. Sheldon D. Goldstein, P.E. Managing Partner, The Steele Group

Using Kruskal-Wallis to Improve Customer Satisfaction. A White Paper by. Sheldon D. Goldstein, P.E. Managing Partner, The Steele Group Using Kruskal-Wallis to Improve Customer Satisfaction A White Paper by Sheldon D. Goldstein, P.E. Managing Partner, The Steele Group Using Kruskal-Wallis to Improve Customer Satisfaction KEYWORDS Kruskal-Wallis

More information

Sample Size and Power in Clinical Trials

Sample Size and Power in Clinical Trials Sample Size and Power in Clinical Trials Version 1.0 May 011 1. Power of a Test. Factors affecting Power 3. Required Sample Size RELATED ISSUES 1. Effect Size. Test Statistics 3. Variation 4. Significance

More information

Simple linear regression

Simple linear regression Simple linear regression Introduction Simple linear regression is a statistical method for obtaining a formula to predict values of one variable from another where there is a causal relationship between

More information

Section 5-3 Binomial Probability Distributions

Section 5-3 Binomial Probability Distributions Section 5-3 Binomial Probability Distributions Key Concept This section presents a basic definition of a binomial distribution along with notation, and methods for finding probability values. Binomial

More information

ESTIMATION - CHAPTER 6 1

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

More information

Comparing Means in Two Populations

Comparing Means in Two Populations Comparing Means in Two Populations Overview The previous section discussed hypothesis testing when sampling from a single population (either a single mean or two means from the same population). Now we

More information

Statistiek I. t-tests. John Nerbonne. CLCG, Rijksuniversiteit Groningen. John Nerbonne 1/35

Statistiek I. t-tests. John Nerbonne. CLCG, Rijksuniversiteit Groningen.  John Nerbonne 1/35 Statistiek I t-tests John Nerbonne CLCG, Rijksuniversiteit Groningen http://wwwletrugnl/nerbonne/teach/statistiek-i/ John Nerbonne 1/35 t-tests To test an average or pair of averages when σ is known, we

More information

Hypothesis Testing: Two Means, Paired Data, Two Proportions

Hypothesis Testing: Two Means, Paired Data, Two Proportions Chapter 10 Hypothesis Testing: Two Means, Paired Data, Two Proportions 10.1 Hypothesis Testing: Two Population Means and Two Population Proportions 1 10.1.1 Student Learning Objectives By the end of this

More information