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# Sociology 601 Class 8: September 24, 2009

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1. Sociology 601 Class 8: September 24, 2009 • 6.6: Small-sample inference for a proportion • 7.1: Large sample comparisons for two independent sample means. • 7.2: Difference between two large sample proportions.

2. 7.1 Large sample comparisons for two independent means • So far, we have been making estimates and inferences about a single sample statistic • Now, we will begin making estimates and inferences for two sample statistics at once. • many real-life problems involve such comparisons • two-group problems often serve as a starting point for more involved statistics, as we shall see in this class.

3. Independent and dependent samples • Two independent random samples: • Two subsamples, each with a mean score for some other variable • example: Comparisons of work hours by race or sex • example: Comparison of earnings by marital status • Two dependent random samples: • Two observations are being compared for each “unit” in the sample • example: before-and-after measurements of the same person at two time points • example: earnings before and after marriage • husband-wife differences

4. Comparison of two large-sample means for independent groups Hypothesis testing as we have done it so far: • Test statistic: z = (Ybar - o) / (s /SQRT(n)) • What can we do when we make inferences about a difference between population means (2 - 1)? • Treat one sample mean as if it were o ? • (NO: too much type I error) • Calculate a confidence interval for each sample mean and see if they overlap? • (NO: too much type II error)

5. Figuring out a test statistic for a comparison of two means Is Y2 –Y1an appropriate way to evaluate 2 - 1? • Answer: Yes. We can appropriately define (2 - 1) as a parameter of interest and estimate it in an unbiased way with (Y2 – Y1) just as we would estimate  with Y. • This line of argument may seem trivial, but it becomes important when we work with variance and standard deviations.

6. Figuring out a standard error for a comparison of two means Comparing standard errors: • A&F 213: formula without derivation • Is s2Ybar2 - s2Ybar1an appropriate way to estimate 2(Ybar2-Ybar1)? • No! • 2(Ybar2-Ybar1)= 2(Ybar2) - 2(Ybar2,Ybar1) + 2(Ybar1) • Where 2(Ybar2,Ybar1) reflects how much the observations for the two groups are dependent. • For independent groups, 2(Ybar2,Ybar1) = 0, so 2(Ybar2-Ybar1)= 2(Ybar2) + 2(Ybar1)

7. Step 1: Significance test for 2 - 1 • The parameter of interest is 2 - 1 • Assumptions: • the sample is drawn from a random sample of some sort, • the parameter of interest is a variable with an interval scale, • the sample size is large enough that the sampling distribution of Ybar2 – Ybar1 is approximately normal. • The two samples are drawn independently

8. Step 2: Significance test for 2 - 1 • The null hypothesis will be that there is no difference between the population means. This means that any difference we observe is due to random chance. • Ho: 2 - 1 = 0 • (We can specify an alpha level now if we want) • Q: Would it matter if we used Ho: 1 - 2 = 0 ? Ho: 1 = 2 ?

9. Step 3: Significance test for 2 - 1 • The test statistic has a standard form: • z = (estimate of parameter – Ho value of parameter) standard error of parameter • Q: If the null hypothesis is that the means are the same, why do we estimate two different standard deviations?

10. Step 4: Significance test for 2 - 1 P-value of calculated z: • Table A • Stata: display 2 * (1 – normal(z) ) • Stata: testi (no data, just parameters) • Stata: ttest (if data file in memory)

11. Step 5: Significance test for 2 - 1 Step 5: Conclusion. • Compare the p-value from step 4 to the alpha level in step 1. If p < α, reject H0If p ≥α, do not reject H0 • State a conclusion about the statistical significance of the test. • Briefly discuss the substantive importance of your findings.

12. Significance test for 2 - 1: Example • Do women spend more time on housework than men? • Data from the 1988 National Survey of Families and Households: • sex sample size mean hours s.d • men 4252 18.1 12.9 • women 6764 32.6 18.2 • The parameter of interest is 2 - 1

13. Significance test for 2 - 1: Example • Assumptions: random sample, interval-scale variable, sample size large enough that the sampling distribution of 2 - 1is approximately normal, independent groups • Hypothesis: Ho: 2 - 1= 0 • Test statistic: z = ((32.6 – 18.1) – 0) / SQRT((12.9)2/4252 + (18.2)2/6764) = 48.8 • p-value: p<.001 • conclusion: • reject H0: these sample differences are very unlikely to occur if men and women do the same number of hours of housework. • furthermore, the observed difference of 14.5 hours per week is a substantively important difference in the amount of housework.

14. Confidence interval for 2 - 1: • housework example with 99% interval: • c.i…. = (32.6 – 18.1) +/- 2.58*( √((12.9)2/4252 + (18.2)2/6764)) = 14.5 +/- 2.58*.30 = 14.5 +/- .8, or (13.7,15.3) • By this analysis, the 99% confidence interval for the difference in housework is 13.7 to 15.3 hours.

15. Stata: Large sample significance test for 2 - 1 • Immediate (no data, just parameters) • ttesti 4252 18.1 12.9 6764 32.6 18.2, unequal • Q: why ttesti with large samples? • For the immediate command, you need the following: • sample size for group 1 (n = 4252) • mean for group 1 • standard deviation for group 1 • sample size for group 2 • mean for group 2 • standard deviation for group 2 • instructions to not assume equal variance (, unequal)

16. Stata: Large sample significance test for 2 - 1, an example . ttesti 4252 18.1 12.9 6764 32.6 18.2, unequal Two-sample t test with unequal variances ------------------------------------------------------------------------------ | Obs Mean Std. Err. Std. Dev. [95% Conf. Interval] ---------+-------------------------------------------------------------------- x | 4252 18.1 .1978304 12.9 17.71215 18.48785 y | 6764 32.6 .221294 18.2 32.16619 33.03381 ---------+-------------------------------------------------------------------- combined | 11016 27.00323 .1697512 17.8166 26.67049 27.33597 ---------+-------------------------------------------------------------------- diff | -14.5 .2968297 -15.08184 -13.91816 ------------------------------------------------------------------------------ Satterthwaite's degrees of freedom: 10858.6 Ho: mean(x) - mean(y) = diff = 0 Ha: diff < 0 Ha: diff != 0 Ha: diff > 0 t = -48.8496 t = -48.8496 t = -48.8496 P < t = 0.0000 P > |t| = 0.0000 P > t = 1.0000

17. Large sample significance test for 2 - 1: command for a data set (#1) . ttest YEARSJOB, by(nonstandard) unequal Two-sample t test with unequal variances ------------------------------------------------------------------------------ Group | Obs Mean Std. Err. Std. Dev. [95% Conf. Interval] ---------+-------------------------------------------------------------------- 0 | 980 9.430612 .2788544 8.729523 8.883391 9.977833 1 | 379 7.907652 .3880947 7.555398 7.144557 8.670747 ---------+-------------------------------------------------------------------- combined | 1359 9.005887 .2290413 8.443521 8.556573 9.4552 ---------+-------------------------------------------------------------------- diff | 1.522961 .4778884 .5848756 2.461045 ------------------------------------------------------------------------------ diff = mean(0) - mean(1) t = 3.1869 Ho: diff = 0 Satterthwaite's degrees of freedom = 787.963 Ha: diff < 0 Ha: diff != 0 Ha: diff > 0 Pr(T < t) = 0.9993 Pr(|T| > |t|) = 0.0015 Pr(T > t) = 0.0007

18. Large sample significance test for 2 - 1: command for a data set (#2) . ttest conrinc if wrkstat==1, by(wrkslf) unequal Two-sample t test with unequal variances ------------------------------------------------------------------------------ Group | Obs Mean Std. Err. Std. Dev. [95% Conf. Interval] ---------+-------------------------------------------------------------------- self-emp | 190 48514.62 2406.263 33168.05 43768.03 53261.2 someone | 1263 34417.11 636.9954 22638 33167.43 35666.8 ---------+-------------------------------------------------------------------- combined | 1453 36260.56 648.5844 24722.9 34988.3 37532.82 ---------+-------------------------------------------------------------------- diff | 14097.5 2489.15 9191.402 19003.6 ------------------------------------------------------------------------------ diff = mean(self-emp) - mean(someone) t = 5.6636 Ho: diff = 0 Satterthwaite's degrees of freedom = 216.259 Ha: diff < 0 Ha: diff != 0 Ha: diff > 0 Pr(T < t) = 1.0000 Pr(|T| > |t|) = 0.0000 Pr(T > t) = 0.0000

19. 7.2: Comparisons of two independent population proportions • In 1982 and 1994, respondents in the General Social Survey were asked: “Do you agree or disagree with this statement? ‘Women should take care of running their homes and leave running the country up to men.’” • Year Agree Disagree Total • 1982 122 223 345 • 1994 268 1632 1900 • Total 390 1855 2245 • Do a formal test to decide whether opinions differed in the two years.

20. Step 1: Significance test for π2 - π1 • The parameter of interest is π2 - π1 • Assumptions: • the sample is drawn from a random sample of some sort, • the parameter of interest is a variable with an interval scale, • the sample size is large enough that the sampling distribution of Pihat2 – Pihat1 is approximately normal. • The two samples are drawn independently

21. Step 2: Significance test for π2 - π1 The null hypothesis will be that there is no difference between the population proportions. This means that any difference we observe is due to random chance. • Ho: π2 - π1 = 0 • (State an alpha here if you want to.)

22. Step 3: Significance test for π2 - π1 The test statistic has a standard form: • z = (estimate of parameter – Ho value of parameter) standard error of parameter • Where pihat is the overall weighted average • This means we are assuming equal variance in the two populations. • Q: why do we use an assumption of equal variance to estimate the standard error for the t-test?

23. Step 4: Significance test for π2 - π1 P-value of calculated z: • Table A, or • Stata: display 2 * (1 – normal(z) ), or • Stata: testi (no data, just parameters) • Stata: ttest (if data file in memory)

24. Step 5: Significance test for π2 - π1 Conclusion: • Compare the p-value from step 4 to the alpha level in step 1. If p < α, reject H0If p ≥α, do not reject H0 • State a conclusion about the statistical significance of the test. • Briefly discuss the substantive importance of your findings.

25. Significance test for π2 - π1: Example • Assumptions: random sample, interval-scale variable, sample size large enough that the sampling distribution of 2 - 1is approximately normal, independent groups • Hypothesis: Ho: π2 - π1= 0 • Test statistic: z = (122/345 – 268/1900) / SQRT[(390/2245)*(1 - 390/2245)*(1/345 + 1/1900)] = 9.59 • p-value: p<<.001 • conclusion: • reject H0: attitudes were clearly different in 1994 than in 1982. • furthermore, the observed difference of .21 is a substantively important change in attitudes.

26. Comparisons of two independent population proportions: Confidence Interval • confidence interval: • Notice that there is no overall weighted average Pihat, as there is in a significance test for proportions. • Instead, we estimate two separate variances from the separate proportions. • Why?

27. STATA: Significance test for π2 - π1:immediate command . prtesti 345 .3536 1900 .1411 • STATA needs the following information: • sample size for group 1 (n = 345) • proportion for group 1 (p = 122/345) • sample size for group 2 (n = 1900) • proportion for group 2 (p = 268/1900)

28. STATA: Significance test for π2 - π1:immediate command . prtesti 345 .3536 1900 .1411 Two-sample test of proportion x: Number of obs = 345 y: Number of obs = 1900 ------------------------------------------------------------------------------ Variable | Mean Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- x | .3536 .0257393 .3031518 .4040482 y | .1411 .0079865 .1254467 .1567533 -------------+---------------------------------------------------------------- diff | .2125 .0269499 .1596791 .2653209 | under Ho: .0221741 9.58 0.000 ------------------------------------------------------------------------------ Ho: proportion(x) - proportion(y) = diff = 0 Ha: diff < 0 Ha: diff != 0 Ha: diff > 0 z = 9.583 z = 9.583 z = 9.583 P < z = 1.0000 P > |z| = 0.0000 P > z = 0.0000 Note the use of one standard error (unequal variance) for the confidence interval, and another (equal variance) for the significance test.

29. STATA command for a data set (#1) . prtest nonstandard if (RACECEN1==1 | RACECEN1==2), by(RACECEN1) Two-sample test of proportion 1: Number of obs = 1389 2: Number of obs = 260 ------------------------------------------------------------------------------ Variable | Mean Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- 1 | .2800576 .0120482 .2564436 .3036716 2 | .3538462 .0296544 .2957247 .4119676 -------------+---------------------------------------------------------------- diff | -.0737886 .0320084 -.1365239 -.0110532 | under Ho: .0307147 -2.40 0.016 ------------------------------------------------------------------------------ diff = prop(1) - prop(2) z = -2.4024 Ho: diff = 0 Ha: diff < 0 Ha: diff != 0 Ha: diff > 0 Pr(Z < z) = 0.0081 Pr(|Z| < |z|) = 0.0163 Pr(Z > z) = 0.9919

30. STATA command for a data set (#1) . gen byte wrkslf0=wrkslf-1 (152 missing values generated) . prtest wrkslf0 if wrkstat==1, by(sex) Two-sample test of proportion male: Number of obs = 874 female: Number of obs = 743 ------------------------------------------------------------------------------ Variable | Mean Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- male | .8272311 .0127876 .8021678 .8522944 female | .9044415 .0107853 .8833027 .9255802 -------------+---------------------------------------------------------------- diff | -.0772103 .0167286 -.1099978 -.0444229 | under Ho: .0171735 -4.50 0.000 ------------------------------------------------------------------------------ diff = prop(male) - prop(female) z = -4.4959 Ho: diff = 0 Ha: diff < 0 Ha: diff != 0 Ha: diff > 0 Pr(Z < z) = 0.0000 Pr(|Z| < |z|) = 0.0000 Pr(Z > z) = 1.0000