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Cal Poly STAT 252 - Comparing Two Means

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1 STAT 252 Handout 6 Winter 2010 Comparing Two Means Today we continue to examine how to compare two groups, but we turn our attention to studies involving a quantitative response variable. Example 1: Got a Tip? Can waitresses increase their tips simply by introducing themselves by name when they greet customers? Garrity and Degelman (1990) report on a study in which a waitress collected data on two-person parties that she waited on during Sunday brunch (with a fixed price of $23.21) at a Charley Brown’s restaurant in southern California. For each party the waitress used a random mechanism to determine whether to give her name as part of her greeting or not. Then she kept track of how much the party gave for a tip at the end of their meal. a) Is this an observational study or a randomized experiment? Explain. b) Identify and classify the explanatory and response variables. c) How do these variables differ from the ones we have studied recently? d) State the null and alternative hypotheses, in symbols and in words, for testing the waitress’ conjecture. Whereas earlier we learned a procedure for assessing whether two proportions differ significantly, today we will study the analogous procedure for comparing two means:  The null hypothesis is H0: μ1 = μ2  The alternative hypothesis is Ha: μ1 > μ2 or Ha: μ1 < μ2 or Ha: μ1 ≠ μ2.  The test statistic is: 22212121nsnsxxt+−=.  The P-value is the area under the t-distribution (with the smaller of n1-1 and n2-1 degrees of freedom) more extreme than the test statistic, in the direction indicated by the alternative hypothesis.  The test decision is to reject H0 whenever p-value < α (significance level).2  A confidence interval for estimating μ1 - μ2 is given by: ()22212121*nsnstxx +±−.  The technical conditions required for this procedure to be valid are that: o the data come from independent random samples or random assignment to groups o the sample sizes are large (>30 in each group) or the populations follow normal distributions The sample mean tip amount for the 20 parties to which the waitress gave her name was $5.44, with a standard deviation of $1.75. These statistics were $3.49 and $1.13, respectively, for the 20 parties to which the waitress did not give her name. e) Use this information to calculate the test statistic and p-value (by hand). f) What test decision would you reach at the α = .05 level? What does this decision mean in the context of this study? g) Do you have enough information to check whether the technical conditions of the two-sample t-test are satisfied here? If so, check them. If not, explain what additional information you would request from the waitress.3 h) Calculate a 95% confidence interval for the difference in population mean tip amounts between the two experimental treatments (giving name or not). Also interpret what the interval means in this context. i) Use Minitab (Stat> Basic Statistics> 2-Sample t…) to confirm your calculations. j) Regardless of whether or not the technical conditions are met, summarize your conclusions from this test. Be sure to comment on issues of causation and generalizability. Example 2: Body Temperatures Recall the data (BodyTemps.mtw) on body temperatures for a sample of healthy adult males and females. a) Examine and comment on comparative dotplots and boxplots of the body temperatures between the two sexes.4 b) Calculate summary statistics from the sample data. Use these to conduct a t-test of whether the data provide strong evidence that the two sexes differ with regard to average body temperature. Include all components of the test, and also check the technical conditions. Summarize your conclusions. c) Use summary statistics from the sample data to construct a 95% confidence interval for the difference in population means. Interpret the interval, paying particular attention to whether it includes the value zero. d) Verify your calculations with Minitab (Stat> Basic statistics> 2-sample t). e) Summarize your conclusions from this analysis.5 Example 3: Fish oil diet A group of fourteen men were randomly divided into two groups: one group went on a diet of fish oil and the other on a diet of regular oil for two weeks. The reduction in diastolic blood pressure (in millimeters of mercury) was recorded for each subject (a negative value means that the person experienced an increase in blood pressure). The data turned out as follows: Fish oil diet: 8, 12, 10, 14, 2, 0, 0 mean: 6.57 std dev: 5.86 Regular oil diet: -6, 0, 1, 2, -3, -4, 2 mean: -1.14 std dev: 3.18 a) Is this an observational study or an experiment? Explain briefly. b) Identify the explanatory and response variables. Also classify each as categorical or quantitative. c) State the appropriate null and alternative hypotheses, in symbols and in words. d) What type of test is called for here? e) Enter the data into Minitab, and calculate the value of the test statistic and p-value. f) Write a sentence explaining what this p-value is the probability of. g) What would the test decision be at the α=.05 significance level? How about at the α=.01 significance level? Relate this to the context of this study.6 h) What is the smallest significance level α at which your decision would be to reject the null hypothesis? i) Check and comment on whether the technical conditions of this test procedure are satisfied. j) Based on the p-value alone, what can you say about a 95% CI for μF - μR? Explain. k) Based on the p -value alone, what can you say about a 99% CI for μF - μR? Explain. Example 4: SAT coaching (hypothetical) Suppose that 5000 students are randomly assigned to either take an SAT coaching course or not, with the following results in their improvements in SAT scores: Sample size Sample mean Sample std dev Coaching group: 2500 46.2 14.4 Control group: 2500 44.4 15.3 a) Use Minitab (Stat> Basic statistics> 2-sample t, summarized data) to conduct a test of whether the sample data provide evidence that SAT coaching is helpful. State the hypotheses, and report the p-value. Draw a conclusion in the context of this study.7 b) Use Minitab to produce a 99% CI for the difference in population mean improvements between the two groups. Interpret this interval. c) Are the test


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Cal Poly STAT 252 - Comparing Two Means

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