DSCI 346 Yamasaki Lecture 2 Hypothesis Tests for Differences in Two Populations

Example of a hypothesis test concerning the difference between 2 independent population means. To determine whether there was sufficient evidence to indicate that the average retirement age in Japan is higher than the average retirement age in the US, the Bureau of Labor Statistics obtains a sample of 30 retired US workers, and 30 retired Japanese workers. The mean and the standard deviation of the retirement ages are as follows: Location US Japan Sample size Sample mean 64.6 yrs 67.5 yrs Sample std dev 4.0 yrs 4.5 yrs 2DSCI 346 Lect 2 (15 pages)

If  1 -  2 = 0, then we would expect x 1 -x 2 would be close to zero and we would reject H 0 if x 1 -x 2 is too big or too small (depending on which way we subtract). How do we determine when x 1 -x 2 is too big or too small? Central Limit Theorem From Central Limit Theorem, we have We don’t have  1 or  2, but since sample sizes are large enough, pretend  1 =s 1 and  2 =s 2. 3DSCI 346 Lect 2 (15 pages)

H 0 :  U =  J (  u -  j = 0) H A :  U <  J (  u -  j < 0) For variety sake, let  =.10 Reject H 0 if z ≤ We could reverse the subtraction and reject if too big DSCI 346 Lect 2 (15 pages)

Central Limit Theorem From Central Limit Theorem we have Under H 0, Since is ≤ , we can reject H 0. We conclude that there is sufficient evidence to indicate that the average retirement age in the US is less than the average retirement age in Japan. Note: If you are doing a one tailed test, you need to be careful of the order  1 -  2 0 5DSCI 346 Lect 2 (15 pages)

Small Sample Hypothesis Testing concerning the difference between two independent means For large samples we had: With small samples we have to assume  1 =  2 and our test statistic becomes: 6DSCI 346 Lect 2 (15 pages)

Example: A retirement investment study is undertaken and the results are summarized below. Test the claim that there is a difference in the mean annual contribution for those covered by TSAs and those with 401(k)s. (  =.05) TSA401(k) n T = 15n 4 = 15 x T = 2,119.70x 4 = 1, s T = s 4 = DSCI 346 Lect 2 (15 pages) H 0 :  T =  4 H 1 :  T ≠  4 df = n 1 +n 2 -2 = = 28 t 28,.025 = 2.048

Since 1.431<2.048, we can’t reject the null hypothesis and there is insufficient evidence to conclude there is a difference in the average annual contribution between those with TSAs ad those with 401(k)s. 8DSCI 346 Lect 2 (15 pages)

Hypothesis Test for differences in two means when the populations are not independent As in the case of confidence intervals we will take differences and treat as a one sample problem with n= number of paired differences. 9DSCI 346 Lect 2 (15 pages)

Example: A hypnotherapist claims that by hypnosis he can increase a sensory measurement score by an average of at least 6. In order to determine if there is sufficient evidence to refute the claim, 8 subjects were tested for sensory measurements before and after hypnosis with the following results: SubjectAfterBefore A B C D E F G H DSCI 346 Lect 2 (15 pages)

SubjectAfterBefore Diff(A-B) A B C D E F G H DSCI 346 Lect 2 (15 pages)

Since < reject null hypothesis, conclude average improvement <6 12DSCI 346 Lect 2 (15 pages)

Hypothesis Test involving the difference of two independent proportions Lets consider two independent populations. We want to test the hypothesis H 0 :  1 =   (lets call this common value  c ) Recall from earlier, using the Central Limit Theorem we have 13DSCI 346 Lect 2 (15 pages)

As before we don’t have what we need to calculate the standard error. We need to estimate it. Let x 1 = number of successes in sample 1 Let x 2 = number of successes in sample 2 Since under the null hypothesis the two proportions are equal, we are going to “pool” our samples together to get an estimate of this common value,  c. DSCI 346 Lect 2 (15 pages)14

Example: A study conducted by the Institute for Journalism Education provided the following results: Of the 175 minority journalists, 23 left journalism. Of the 125 non-minority journalists, 6 had left journalism. Do these data provide sufficient evidence to indicate the proportion of minorities leaving the journalism profession is higher than non- minorities? (  =.05) Since > there is sufficient evidence to conclude that a significantly higher proportion of minority journalists leave journalism 15DSCI 346 Lect 2 (15 pages)