Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 24- 1

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 24 Comparing Means

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Plot the Data The natural display for comparing two groups is boxplots of the data for the two groups, placed side- by-side. For example:

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Comparing Two Means Once we have examined the side-by-side boxplots, we can turn to the comparison of two means. Comparing two means is not very different from comparing two proportions. This time the parameter of interest is the difference between the two means,  1 –  2.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Comparing Two Means (cont.) Remember that, for independent random quantities, variances add. So, the standard deviation of the difference between two sample means is We still don’t know the true standard deviations of the two groups, so we need to estimate and use the standard error

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Comparing Two Means (cont.) Because we are working with means and estimating the standard error of their difference using the data, we shouldn’t be surprised that the sampling model is a Student’s t. The confidence interval we build is called a two-sample t-interval (for the difference in means). The corresponding hypothesis test is called a two-sample hypothesis test.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Sampling Distribution for the Difference Between Two Means When the conditions are met, the standardized sample difference between the means of two independent groups can be modeled by a Student’s t-model with a number of degrees of freedom found with a special formula. We estimate the standard error with

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Assumptions and Conditions Independence Assumption (Each condition needs to be checked for both groups.): Randomization Condition: Were the data collected with suitable randomization (representative random samples or a randomized experiment)? 10% Condition: We don’t usually check this condition for differences of means. We will check it for means only if we have a very small population or an extremely large sample.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Assumptions and Conditions (cont.) Normal Population Assumption: Nearly Normal Condition: This must be checked for both groups. A violation by either one violates the condition. Independent Groups Assumption: The two groups we are comparing must be independent of each other. (See Chapter 25 if the groups are not independent of one another…)

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Two-Sample t-Interval When the conditions are met, we are ready to find the confidence interval for the difference between means of two independent groups. The confidence interval is where the standard error of the difference of the means is The critical value depends on the particular confidence level, C, that you specify and on the number of degrees of freedom, which we get from the sample sizes and a special formula.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Degrees of Freedom The special formula for the degrees of freedom for our t critical value is a bear: Because of this, we will let technology calculate degrees of freedom for us!

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Testing the Difference Between Two Means The hypothesis test we use is the two-sample t- test for means. The conditions for the two-sample t-test for the difference between the means of two independent groups are the same as for the two- sample t-interval.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Testing the Difference Between Two Means (cont.) We test the hypothesis H 0 :  1 –  2 =  0, where the hypothesized difference,  0, is almost always 0, using the statistic The standard error is When the conditions are met and the null hypothesis is true, this statistic can be closely modeled by a Student’s t-model with a number of degrees of freedom given by a special formula. We use that model to obtain a P-value.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example 1: Pulse Rates Resting pulse rates for a random sample of 26 smokers had a mean of 80 beats per minute (bpm) and a standard deviation of 5 bpm. Among 32 randomly chosen nonsmokers, the mean and standard deviation were 74 and 6 bpm. Both sets of data were roughly symmetric and had no outliers. a. Create a 95% confidence interval for the difference in the resting pulse rates between smokers and nonsmokers. b. Is there evidence of a difference in mean pulse rate between smokers and non-smokers? How big?

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example 1 continued…. Hypothesis: The null hypothesis is that there is no difference between the pulse rates of smokers and non- smokers. The alternative hypothesis is that there is a difference between the resting pulse rates between smokers and non-smokers. μ s =population mean for pulse rate for smokers μ n =population mean for pulse rate for non-smokers

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example 1… 1. The samples were random, as stated in the problem. 2. The pulse rate of one person (smoker or non-smoker) does not affect the pulse rate of another person smokers is less than 10% of all smokers. 32 is less than 10% of all non-smokers. 4. Nearly Normal: Graph boxplot of both. Both graphs seem unimodal and approximately symmetric, so the both sampling distributions are nearly Normal. 5. I assume that two groups were chosen independently of each other. Since all conditions are met, we can use the 2 sample t-test.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example 1… Mechanics:

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example 1… Conclusion: Using α=.05 and the 2 sample t-test, the p-value was.0001 which is much lower than α. We reject the null hypothesis. We have strong evidence to suggest a difference in mean pulse rates for smokers and non-smokers. Using a 2 sample t interval, I am 95% confident that the difference in average pulse rate for smokers is between 3.1 and 8.9 beats per minute higher than for nonsmokers. (3.1,8.29)

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example 2: Pizza!!! Here are the saturated fat content (in grams) for several pizzas sold by two national chains. a. Create a 90% confidence interval for the difference in fat content in pizza sold by the two chains. b. We want to know if Dominos has significantly higher mean saturated fat content than Papa John’s pizza. Dominos Papa John’s

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example 2 continued… 1. There is no mention of randomness in the problem. I will assume that these pizzas are representative of the pizzas sold by these two chains. 2. The fat content of one pizza (either chain) does not influence the fat content of another pizza pizzas is less than 10% of all Domino’s pizza. 15 is less than 10% of all Papa John’s pizza. 4. Nearly Normal: Graph boxplot of both. Both graphs are roughly unimodal and approximately symmetric, so the both sampling distributions are nearly Normal. Since all conditions are met, we can use the 2 sample t-test.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example 2 continued… Hypothesis: The null hypothesis is that there is no difference between the saturated fat content of Dominos and Papa John’s pizza. The alternative hypothesis is that Domino’s pizza has a higher saturated fat content than Papa John’s. μ d =population mean for saturated fat for Dominos pizza μ p =population mean for saturated fat for Papa John’s pizza

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example 2 continued… Mechanics:

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example 2 completed. Conclusion: Using α=.05 and the 2 sample t-test, the p-value was which is much lower than α. We reject the null hypothesis. We have strong evidence to suggest that Dominos pizza has a higher saturated fat content than Papa John’s. Using a 2 sample t-interval, I am 90% confident that the difference in the average saturated fat content between Domino’s Pizza is between and higher than Papa John’s pizza.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Back Into the Pool Remember that when we know a proportion, we know its standard deviation. Thus, when testing the null hypothesis that two proportions were equal, we could assume their variances were equal as well. This led us to pool our data for the hypothesis test.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Back Into the Pool (cont.) For means, there is also a pooled t-test. Like the two-proportions z-test, this test assumes that the variances in the two groups are equal. But, be careful, there is no link between a mean and its standard deviation…

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Back Into the Pool (cont.) If we are willing to assume that the variances of two means are equal, we can pool the data from two groups to estimate the common variance and make the degrees of freedom formula much simpler. We are still estimating the pooled standard deviation from the data, so we use Student’s t- model, and the test is called a pooled t-test.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide *The Pooled t-Test If we assume that the variances are equal, we can estimate the common variance from the numbers we already have: Substituting into our standard error formula, we get: Our degrees of freedom are now df = n 1 + n 2 – 2.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide *The Pooled t-Test and Confidence Interval The conditions for the pooled t-test and corresponding confidence interval are the same as for our earlier two- sample t procedures, with the assumption that the variances of the two groups are the same. For the hypothesis test, our test statistic is which has df = n 1 + n 2 – 2. Our confidence interval is

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Is the Pool All Wet? So, when should you use pooled-t methods rather than two-sample t methods? Never. (Well, hardly ever.) Because the advantages of pooling are small, and you are allowed to pool only rarely (when the equal variance assumption is met), don’t. It’s never wrong not to pool.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Why Not Test the Assumption That the Variances Are Equal? There is a hypothesis test that would do this. But, it is very sensitive to failures of the assumptions and works poorly for small sample sizes—just the situation in which we might care about a difference in the methods. So, the test does not work when we would need it to.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Is There Ever a Time When Assuming Equal Variances Makes Sense? Yes. In a randomized comparative experiment, we start by assigning our experimental units to treatments at random. Each treatment group therefore begins with the same population variance. In this case assuming the variances are equal is still an assumption, and there are conditions that need to be checked, but at least it’s a plausible assumption.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide *Tukey’s Quick Test To use Tukey’s test, one group must have the highest value and the other, the lowest.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide *Tukey’s Quick Test (cont.) Count how many values in the high group are higher than all the values in the lower group. Add to this the number of values in the low group that are lower than all the values in the higher group. (Count ties as ½.) If this total is 7 or more, we can reject the null hypothesis of equal means at  = The “critical values” of 10 and 13 give us  ’s of 0.01 and

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide *Tukey’s Quick Test (cont.) This is another example of a distribution-free test and a remarkably good test. The only assumption it requires is that the two samples be independent. Since this test is simple, consider using it as a check of your two-sample t results—if the results disagree, check the assumptions.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide What Can Go Wrong? Watch out for paired data. The Independent Groups Assumption deserves special attention. If the samples are not independent, you can’t use two-sample methods. Look at the plots. Check for outliers and non-normal distributions by making and examining boxplots.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide What have we learned? We’ve learned to use statistical inference to compare the means of two independent groups. We use t-models for the methods in this chapter. It is still important to check conditions to see if our assumptions are reasonable. The standard error for the difference in sample means depends on believing that our data come from independent groups, but pooling is not the best choice here. The reasoning of statistical inference remains the same; only the mechanics change.