Chapter 24 Comparing Means Copyright © 2009 Pearson Education, Inc.

Objectives: The student will be able to: Perform and interpret a two-sample t-test for two population means, to include: writing appropriate hypotheses, checking the necessary assumptions, drawing an appropriate diagram, computing the P-value, making a decision, and interpreting the results in the context of the problem. Compute and interpret in context a t-based confidence interval for the difference between two population means, checking the necessary assumptions.

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:

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.

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 t-test.

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.

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…)

To find the confidence interval for the difference of two population means – 2-SampTInt In the morning class, the mean on the first exam was 78 with a standard deviation of 2.3 for the 57 students in the class. In the afternoon class, the mean was 81 with a standard deviation of 5.7 for the 37 students in the class. Determine the 90% confidence interval for the difference in the means of the two groups. Stat... Tests.. #0 for 2-Samp T Int... enter For Inpt choose Stats unless you have placed the data in L1 and L2. X bar 1 is 78 and sx1 is 2.3 with n1 as 57 Xbar2 is 81 and sx2 is 5.7 with n2 as 37 C-level is .9 Never pool Calculate You should get an interval from -4.656 to -1.344 indicating that there is a difference between the two means of 1.344 to 4.656 points. (Don’t report a negative difference.)

15. The data below show the sugar content (as a percentage of weight) of several national brands of children’s and adult’s cereals. Create and interpret a 95% confidence interval for the difference between in mean sugar content. Be sure to check the necessary assumptions and conditions Children’s: 40.3 55 45.7 43.3 50.3 45.9 53.5 43 44.2 44 47.4 44 33.6 55.1 48.8 50.4 37.8 60.3 46.6 Adults’: 20 30.2 2.2 7.5 4.4 22.2 16.6 14.5 21.4 3.3 6.6 7.8 10.6 16.2 14.5 4.1 15.8 4.1 2.4 3.5 8.5 10 1 4.4 1.3 8.1 4.7 18.4

FYI -- 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

FYI --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.

FYI -- 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!

Testing the Difference Between Two Means The hypothesis test we use is the two-sample t-test for the difference between 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.

FYI – A Test for the Difference Between Two Means We test the hypothesis H0: 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.

Using the TI to do a hypothesis test for difference of the means, independent samples. It has been suggested that college students learn more and obtain higher grades in small classes (40 or less) when compared to large classes (150 or more). To test this claim, a university assigned a professor to teach a small and a large class of the same course. At the end of the course, the classes were given the same exam. The following are the final grade results. Test the claim at an alpha of 0.05. Sample size              35                    170 Sample mean           74.2                  71.7 Standard deviation      14                   13  H0: μsmall=μlarge  HA: μsmall>μlarge  Stat…Test…2-sampT-test  and select STATS not data fill in the appropriate statistics For μ1: choose > don't pool Calculate The p value is .1676 which causes us NOT to reject the null hypothesis which means the smaller class did not do better.

29. A study was conducted to assess the effects that occur when children are exposed to cocaine before birth. 190 children born to cocaine users had a mean score of 7.3 (with a standard deviation of 3.0) on a certain aptitude test. 186 children not exposed to cocaine had a mean score of 8.2 with a standard deviation of 3.0. Use an alpha of 0.05 to test the claim that cocaine use is harmful to children’s aptitude (Triola 2008). H0: µ1=µ2 HA: µ1<µ2 Test statistic (t) = -2.91 P-value = 0.002 Conclusion: Reject the null. We can conclude with reasonable certainty that cocaine is bad for children. Moral: Don’t use cocaine when you are pregnant.

Having done poorly on their math final exams in June, six students repeat the course in summer school, then take another exam in August. If we consider these students representative of all students who might attend this summer school in other years, do these results provide evidence that the program is worthwhile? June: 54, 49, 68, 66, 62, 62 August: 50, 65, 74, 64, 68, 72

Small bowl: n: 26, y(bar): 5.07oz, s: 1.84oz Researchers investigated how the size of a bowl affects how much ice cream people tend to scoop when serving themselves. At an “ice cream social”, people were randomly given either a 17 oz or a 34 oz bowl and were invited to scoop as much ice cream as they liked. Did the bowl size change the selected portion size? Small bowl: n: 26, y(bar): 5.07oz, s: 1.84oz Large bowl: n: 22, y(bar): 6.58oz, s: 2.91oz Test an appropriate hypothesis and state your conclusions (for assumptions and conditions that you cannot test, assume they are sufficiently satisfied to proceed)

Is it a good idea to listen to music when studying for a test Is it a good idea to listen to music when studying for a test? In a study conducted by some statistics students, 62 people were randomly assigned to listen to rap music, Mozart, or no music while attempting to memorize objects pictured on a page. They were then asked to list all the objects they could remember. Here are summary statistics for each group: Rap: n: 29, mean: 10.72, SD: 3.99 Mozart: n: 20, mean: 10.00, SD: 3.19 No music: n: 13, mean: 12.77, SD: 4.73 Does it appear that it is better to study while listening to Mozart than to rap music? Test an appropriate hypothesis and state your conclusion Create a 90% confidence interval for the mean difference between students who study to Mozart and those who listen to no music at all. Interpret your interval.

FYI – Why we say No to Pooled 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. For means, 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 (for the difference between means). 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.

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.

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.

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.