1 1 Chapter 2: Comparing Means 2.1 One-Sample t -Test 2.2 Paired t -Test 2.3 Two-Sample t -Test

2 2 Chapter 2: Comparing Means 2.1 One-Sample t -Test 2.2 Paired t -Test 2.3 Two-Sample t -Test

3 Objectives Use JMP to evaluate the t-test assumption of normality. Construct lower-tailed, upper-tailed, and two-tailed hypothesis tests using the t-test. Identify the appropriate test statistic and hypothesis test to answer a research question. 3

4 One-Sample t -Test Assumptions The one-sample t-test assumes that is normally distributed. Normal populations yield normal sample means. –Decide whether population is normal based on tests of sample data. –Small samples might weakly support these tests. Non-normal populations might yield normal sample means. –Large samples benefit from principle of central limit theorem. 4

5 Evaluating Normality In addition to graphical tools, JMP provides a statistical test for normality with the following hypotheses: H 0 :the data come from a normal distribution H 1 : the data do not come from a normal distribution 5

6 Statistical Tests for Normality Three tests are provided by the Distribution platform to assess the normality of the data. Shapiro-Wilk test for up to 2000 observations. Kolmogorov-Smirnov-Lilliefors (KSL) test for more than 2000 observations. Lilliefors confidence region in the normal quantile plot. 6

7 One-Sample t -Test Assumptions The one-sample t-test assumes that observations are independent. Independence is determined by the data collection method. An appropriate simple random sample should lead to independent observations. Advanced methods handle restrictions to the randomization both in the design and in the analysis of data from an experiment. 7

8 Three Research Problems An electronics manufacturer wants to determine whether, on average, its power supplies are supplying significantly more or less than the desired 12 amperes. An agricultural company is confident that its new fertilizer results in crop yields of more than 55 bushels per acre, on average. A laboratory group is concerned that the purity of a chemical that it is using is less than 99.4% on average, as the supplier claims. 8

9 How to Choose the Appropriate t -Test Examine the alternative hypothesis to determine what t-test is appropriate. Use a two-tailed test if you suspect the true mean is different than claimed (H 1 :   12). Use an upper-tailed test if you suspect that the true mean is higher than claimed (H 1 :  > 55 ). Use a lower-tailed test if you suspect that the true mean is lower than claimed (H 1 :  < 99.4). 9

Multiple Answer Poll The null hypothesis is which of the following? (Select all of the correct answers.) a.The logical opposite of the alternative hypothesis b.The claim that you suspect is true c.Always assumed to be true d.Your instinct 10

Multiple Answer Poll – Correct Answers The null hypothesis is which of the following? (Select all of the correct answers.) a.The logical opposite of the alternative hypothesis b.The claim that you suspect is true c.Always assumed to be true d.Your instinct 11

12 Three t -Tests and Their p -Values 12 Two-tailed ( |t| ) Upper-tailed ( > ) Lower-tailed ( < )

Multiple Choice Poll A professor suspects her class is performing below the department average of 73%. She decides to test this claim. Which of the following is the correct alternative hypothesis? a. μ <.73 b. μ >.73 c. μ ≠.73 13

Multiple Choice Poll – Correct Answer A professor suspects her class is performing below the department average of 73%. She decides to test this claim. Which of the following is the correct alternative hypothesis? a. μ <.73 b. μ >.73 c. μ ≠.73 14

15 The Test Statistic The statistic follows a t-distribution with mean zero and a standard deviation that depends on the degrees of freedom. 15 where µ 0 is the hypothesized mean. is the sample mean. is the estimated standard error of the mean.

16 Research Question A research company suspects that drivers are going faster than the posted speed limit of 45 mph. Help it test the following hypotheses: 16 H 0 :  ≤ 45 H 1 :  > 45  = 0.05

17 The Reference Distribution Compute 17 for your sample and compare the sample t to the reference distribution.

18 Analysis Workflow Follow these steps in your analysis: Plot your data Diagnose data problems Perform your analysis / Choose your model Evaluate model assumptions Diagnose statistics or model problems Iterate as necessary 18

19 This demonstration illustrates the concepts discussed previously. Upper-Tailed Hypothesis Test

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21 Setup for the Poll Examine the Goodness-of-Fit Test output from the demonstration. 21

Poll At α =.05, is there sufficient evidence to reject normality of the data?  Yes  No 22

Poll – Correct Answer At α =.05, is there sufficient evidence to reject normality of the data?  Yes  No 23

24 This exercise reinforces the concepts discussed previously. Exercise

Multiple Choice Poll In the exercise, a one-sample t-test examined whether the average weight of cereal boxes was significantly different from 15. What type of test was run? a.An upper-tailed test b.A lower-tailed test c.A two-sided test 25

Multiple Choice Poll – Correct Answer In the exercise, a one-sample t-test examined whether the average weight of cereal boxes was significantly different from 15. What type of test was run? a.An upper-tailed test b.A lower-tailed test c.A two-sided test 26

27 Chapter 2: Comparing Means 2.1 One-Sample t -Test 2.2 Paired t -Test 2.3 Two-Sample t -Test

28 Objectives Recognize when a paired t-test is appropriate. Perform a paired t-test. 28

29 Paired t -Test A paired t-test is a one-sample t-test on the mean of the differences between paired observations. A paired t-test tests the following hypotheses: where d is the difference between paired measurements. 29 H 0 : µ d = 0 H 1 : µ d  0

30 The Test Statistic Use this test statistic to determine whether two means from paired samples differ: where is the mean difference between the paired observations (d i = y 2i – y 1i ). is the hypothesized mean difference (H 0 : µ d = 0). is the standard error of the mean difference. 30

31 Paired t -Test Assumptions The assumptions of the paired t-test are as follows: is normally distributed measurements among subjects are independent the difference should not be proportional to the response 31

32 Total Cholesterol Example A clinical chemist is interested in whether cholesterol measurements are different for pairs of fresh and frozen serum samples obtained from a group of patients. 32

33 This demonstration illustrates the concepts discussed previously. Paired t -Test

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Multiple Answer Poll Which of the following statements are true? a.A paired t-test is the same as a one-sample t-test on the differences. b.The Matched Pairs platform in JMP allows an evaluation of the relationship, if any, between the differences and the response level. c.A paired t-test is used when pairs of observations are related. 35

Multiple Answer Poll – Correct Answers Which of the following statements are true? a.A paired t-test is the same as a one-sample t-test on the differences. b.The Matched Pairs platform in JMP allows an evaluation of the relationship, if any, between the differences and the response level. c.A paired t-test is used when pairs of observations are related. 36

37 This exercise reinforces the concepts discussed previously. Exercise

Quiz Examine the output from the exercise. What is the conclusion from these comparisons of blood pressure before and after the training? 38

Quiz – Correct Answer Examine the output from the exercise. What is the conclusion from these comparisons of blood pressure before and after the training? The 95% confidence interval on the true average does not contain zero in either case, so this indicates that there is a difference, on average, between the blood pressure readings before and after the training. 39

40 Chapter 2: Comparing Means 2.1 One-Sample t -Test 2.2 Paired t -Test 2.3 Two-Sample t -Test

41 Objectives Distinguish between a two-sample t-test and a paired t-test. Identify the three assumptions of a two-sample t-test. Compare the means of two populations using a two-sample t-test. 41

42 Two-Sample t -Test or Paired t -Test? A two-sample t-test, or independent t-test, is used when you compare the means of two independent populations. A paired t-test is used when there are pairs of related measurements. 42

Multiple Choice Poll Suppose that you are interested in evaluating the effect of a marketing campaign on regional retail sales. You collect data before and after the campaign for each retail facility. Which is the more appropriate test? a.One-sample t-test b.Paired t-test c.Two-sample t-test d.Any of these tests could be appropriate. 43

Multiple Choice Poll – Correct Answer Suppose that you are interested in evaluating the effect of a marketing campaign on regional retail sales. You collect data before and after the campaign for each retail facility. Which is the more appropriate test? a.One-sample t-test b.Paired t-test c.Two-sample t-test d.Any of these tests could be appropriate. 44

45 Concrete Example 45 ReinforcedStandard

46 Statistical Assumptions Independent observations is normally distributed for each population Equal variances for each population 46 µ1µ1 µ2µ2

47 The Test Statistic Use this t-test to test whether the means of two independent groups are different: 47 where is the test statistic. is the mean for the sample from population i (i=1, 2). is the pooled standard error.

48 Example of a Two-Sided Hypothesis Suppose you are interested in determining whether two group means are different from one another. The null and alternative hypotheses are the following: H 0 : the group means are identical (  1 =  2 ) H 1 : the group means are different (  1   2 ) 48

49 This demonstration illustrates the concepts discussed previously. Comparing Two Means Using a t -Test

50 Setup for the Poll Examine the output of the tests of equal variances. 50

Poll At α =.05, is there sufficient evidence to reject equal variances? Hint: recall that H 0 : σ 1 2 = σ 2 2 and H 1 : σ 1 2 ≠ σ 2 2  Yes  No 51

Poll – Correct Answer At α =.05, is there sufficient evidence to reject equal variances? Hint: recall that H 0 : σ 1 2 = σ 2 2 and H 1 : σ 1 2 ≠ σ 2 2  Yes  No 52

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54 This exercise reinforces the concepts discussed previously. Exercise

55 Setup for the Poll Examine the output from the exercise to determine which p-value would be appropriate if someone were to test whether the risks at public hospitals were lower than private hospitals. 55

Multiple Choice Poll Which p-value would be appropriate if someone were to test if the risks at public hospitals were lower than private hospitals? a b c

Multiple Choice Poll – Correct Answer Which p-value would be appropriate if someone were to test if the risks at public hospitals were lower than private hospitals? a b c