Chapter 19 main topics. Sociology 360 Statistics for Sociologists I Chapter 19 Two-Sample Problems. Chapter 19 homework assignment

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1 Sociology 360 Statistics for Sociologists I Chapter 19 Two-Sample Problems Chapter 19 main topics Two-sample t procedures Robustness of two-sample t procedures Details of the t approximation Avoid the pooled two-sample t procedures Avoid inference about standard deviations Topic to omit: The F test for comparing two standard deviations 1 Chapter 19 homework assignment Problems: 19.6,.7,.9,.14,.16,.30,.3 One- vs. two-sample t-tests One-sample test: Is a population mean >, <, or different from some fixed value? Two-sample test: Goal: Compare responses to two treatments or characteristics of two populations. Independent samples for each treatment or population (i.e., the data are not matched pairs). Are the population means the same as each other, or is one greater than the other? 3 4

2 Examples of two-sample tests Research questions: Do men have higher salaries than women? Where do people travel farther to work, Detroit or Los Angeles? Two-sample t-test: assumptions We have an SRS from each of two populations or an experiment with two randomly assigned groups. We can consider subgroups of individuals in a random sample (e.g., men, women) as independent samples from their respective populations. The samples are independent. The individuals that make up the two samples are not related to each other (cannot be paired or matched). If cases in the two samples can be paired or matched, used a matched pairs design. 5 6 Problems: Identify the appropriate method Is this a one-sample, two-sample, one-sample matched pairs problem? Would you perform a hypothesis test or find a confidence interval? 1. Do Burger King Whoppers have more than 670 calories?. Do Whoppers have more calories than Big Macs? 3. Mothers of twins were surveyed and asked how often in the past month strangers had asked whether the twins were identical. 4. Are parents equally strict with boys and girls? In a random sample of families, researchers asked a brother and sister from each family to rate how strict their parents were. Null hypothesis for a two-sample test Most frequently, the null hypothesis is that the two means are the same. Optio: H0:!1 =! Option : H0:!1 -! = 0 Of course both options mean the same thing since Option is obtained algebraically from Optio by subtracting! from both sides. Option, however, in stating that the difference between the two population means is zero, focusses our attention on the proper sample statistic for inference, the difference in sample means: ( x 1 x ) 7 8

3 Possible alternative hypotheses Two-tailed: Optio: H A :! 1!! or Option : H A :! 1 -!! 0 One-tailed (right): Optio: H A :! 1 >! or Option : H A :! 1 -! > 0 One-tailed (left) Optio: H A :! 1 <! or Option : H A :! 1 -! < 0 In each case, Optio is equivalent to Option. Whoppers and Big Macs Do Whoppers have more calories than Big Macs? Let! w = Mean calories in Whoppers Let! bm = Mean calories in Big Macs Write the null and alternative hypotheses using both methods (optio and option ) 9 10 Sampling distribution of the difference in means Our interest centers on the difference between the two population means,!1 -!, which I will emphasize is a single numerical value by writing it within parentheses, like this: (!1 -!). We can estimate (!1 -!) by its sample analog, ( x 1 x ). Since ( x 1 x ) is a number calculated only from sample information, it is a statistic. As a statistic, ( x 1 x ) has a sampling distribution. The sampling distribution of ( x 1 x ) will be Normal under the right circumstances. And the mean of that sampling distribution will be (!1 -!). All that remains to be discovered about the sampling distribution is its standard error (or estimated standard deviation). Sampling distribution of the difference in means µ G µ B =

4 Standard error The two-sample t statistic follows approximately the t distribution with a standard error SE reflecting variation from both samples. In fact, its standard error is simply the square root of the sum of the standard errors of each sample considered separately: s 1 SE = + s n df Degrees of freedom Since we are using a standard error, estimated from the data, rather than a known standard deviation, the procedures will be t rather than z based. That means we need to have a value for the degrees of freedom of the t distribution. A conservative approach is to use the smaller of (n1-1) and (n - 1) as the degrees of freedom. This rule is conservative in that it may give a value larger than is really appropriate, which leads to wider confidence intervals and larger P-values (meaning we are a bit less likely to reject H0). You should use this rule for problems done by hand; for example, on the exam. µ 1 "µ Two-sample t-test The null hypothesis is that both population means! 1 and! are equal, thus their difference is equal to zero: H 0 : (µ 1 µ ) = 0 with either a one-sided or a two-sided alternative hypothesis. We construct a t statistic via the usual comparison of the observed statistic to the hypothesized value: t = ( x 1 x ) (µ 1 µ ) 0 SE = ( x 1 x ) 0 s 1 + s n Ideal number of children Do men and women have different beliefs about the ideal number of children in a family? 004 General Social Survey asked, What do you think is the ideal number of children for a family to have? Here is a summary of the responses: Gender x s n Male Female This statistic has an approximate t distribution if H0 is true

5 Ideal number of children Gender x s n Male Female What are the null and alternative hypotheses? Choose an # level. Draw a picture of the sampling distribution and the p-value you are looking for. Perform the test and evaluate the result..89 Note: =.064 Confidence interval As before, we often supplement a hypothesis test by a CI. (And sometimes we omit the test.) For two-sample problems, the question is to estimate the mean of the distribution of the difference scores in the population. The statistic continues to be ( x 1 x ) and the confidence interval is CI = ( x 1 x ) ±t s 1 + s n % CI for the example Gender x s n Male Female df = min(373,416) = 373 For C =.95, t 373 z = Effects of Reading Program on Reading Comprehension New reading activities for elementary school children RA 3 rd graders to treatment group and control group Compare reading comprehension s CI = ( x 1 x ) ±t 1 + s n Note: =.064 Calculate a 95% CI for the effect of the new reading activities on reading comprehension Note: =

6 Robustness of the two-sample t procedures We must have an SRS or randomized comparative experiment. t procedures are only exact if the population distribution is exactly normal. But, we will consider two-sample t procedures good enough approximations in these cases: 1. When n1 + n < 15, the data from both samples must be close to normal (roughly symmetric, single peak) and without outliers.. Whe5 " n1 + n < 40, mild skewness is acceptable, but not outliers. 3. When n1 + n " 40, the t statistic will be valid even with strong skewness. Details of the t approximation The actual distribution of the two-sample t statistic is not really t (!). But it is a distribution that can be very closely approximated by a t distribution with this number of degrees of freedom: df = ( s 1 ) + s n ( ) 1 s n 1 This is known as the Satterthwaite approximation. The formula typically produces a non-integer degree of freedom value. Computers routinely calculate this approximation. You should recognize it when you see it. ( s n ) But on exams, use the smaller of (n1-1) and (n - 1) instead. 1 Avoid the pooled two-sample t procedures Your textbook s author, Moore, recommends completely avoiding the pooled two-sample t procedures, and I agree. Pooled procedures are often the default choice in stat packages (e.g., Stata, including the current version, 10.0). The reasons that the pooled approach is often used are: 1) it was historically easier to calculate; ) it leads to a smaller estimated standard error when the assumptions are met; 3) it amounts to a special case of a very important technique called the analysis of variance. But Moore is right to emphasize: 1) the assumption of normality and equal variances can t be tested effectively when the sample sizes are small (i.e., when the pooled procedure would be most advantageous); ) the pooled procedure can lead to incorrect inferences when the assumptions aren t met; 3) the reduction in SE s is small for large n s. So you are asked to know not to accept a default assumption of equal (pooled) variances, and why not! Avoid inference about standard deviations In an extension of the ideas behind not using the pooled t procedures, Moore also warns us not to try to make inferences about standard deviations at all, at least in smaller samples, and at least without expert statistical help. The problem is that it is hard to make a useful test of the hypothesis that the standard deviations in two populations are the same unless we are willing to assume the shapes of the two distributions are the same. (Things are even easier if we assume the shapes are normal.) But when the sample is small there is no easy way to tell if the shapes of two distributions are the same. So, says Moore, avoid testing of hypotheses that standard deviations are the same. My only reservation about this recommendation would be in cases where there are strong reasons to expect normality in both populations. 3 4

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