Statistics for Social and Behavioral Sciences Session #19: Estimation and Hypothesis Testing, Wrap-up & p-value (Agresti and Finlay, from Chapter 5 to Chapter 6) Prof. Amine Ouazad

Outline 1.True or False? 1.The p-value 2.One sided t tests 3.Type II error Next time:Mid term covering Chapters 1—6 and 9 of A&F

True or False ?? For the test of a null hypothesis, the confidence interval method can only be applied when sample size N is large. For the test of a null hypothesis, the t test method can only be applied when the sample size is small. The sampling distribution of a t statistic is left-skewed, or right skewed, depending on the number of degrees of freedom. When testing a null hypothesis with 95% confidence level, the probability of a Type I error (i.e. when the null hypothesis is true) is 5%.

Outline 1.True or False? 1.The p-value 2.One sided t tests 3.Type II error Next time:Mid term covering Chapters 1—6 and 9 of A&F

The p-value is the probability (in other potential samples, unobserved) that the absolute value of the t statistic is greater than the observed t statistic in our sample. A low p value means….. – that larger absolute values of the t test are unlikely. In practice: Reject the null hypothesis H0 at 95% if the p- value is lower than Similarly for 90% and 99%! (0.10 and 0.01 resp.)

Back to Amartya Sen: Are there really fewer women per man in India? “More than 100 million women are missing” Dataset collected by Simple Random Sampling. Population: Indian residents. Sample: 2,878,380 respondents. Little response bias. Non response bias is possible. Thomas W. Lamont University Professor, and Professor of Economics and Philosophy, at Harvard University

Back to Amartya Sen: Are there really fewer women per man in India? Obtaining Table 5.1’s t scores values in Stata: type `display invttail(df,0.025)’ for 95%. Replace df by the degrees of freedom (N-1)

Either with the confidence interval method, Or with the t test method, We reject the null hypothesis: “H0: the fraction of women in India is 50%” At 99% (and thus at 95%, 90%). Draw the p-value on a graph of the distribution. Back to Amartya Sen: Are there really fewer women per men in India?

Outline 1.True or False? 1.The p-value 2.One sided t tests 3.Type II error Next time:Mid term covering Chapters 1—6 and 9 of A&F

Back to Cory Gardner With 1850 respondents, the polling company PPP found that 48% of respondents would vote for Gardner. Pollsters think that Gardner wins if the vote share is > 46%. Can you test the null hypothesis that the true vote share is = 46%, with H a : vote share > 46% ?

One sided test of H 0 :  =v, H a :  >=v When testing  =v vs  >=v, use the t statistic method. – avoid the confidence interval method. Assume that  =v and build t = (m-v)/SE. ☞ Reject the null hypothesis at 95% if the t statistic is greater than +t ☞ Do not reject the null if t < 0. Similarly for 90% and 99% (t 0.10 and t 0.01 ).

Can we reject the null hypothesis at 95% with a one sided test? Can we reject the null hypothesis at 95% with a two sided test? ☞ Beware of the dangers of a one-sided test !! Back to Cory Gardner

Why it is objectionable to use one-sided tests…. Assuming that the parameter is higher than a certain value…. – Lowers the threshold t necessary to reject the null hypothesis. – For a one sided test, we use t 0.05 for a test at 95% – For a two sided test, we use t for a test at 95%. Ask yourself: Can I assume that the  will be necessarily higher than v?

But here, not such a big issue: Are there really fewer women per man in India? Obtaining one sided t scores values in Stata: type `display invttail(df,0.05)’ for 95%. Replace df by the degrees of freedom (N-1)

Outline 1.True or False? 1.The p-value 2.One sided t tests 3.Type II error Next time:Mid term covering Chapters 1—6 and 9 of A&F

Type II error Type I error is well-known. But what is the level of type II error?

Wrap up Confidence interval method for the test of H 0 :  = v. H a :  ≠ v. CI = [ m – t * SE ; m + t * SE ] – Reject the H 0 with significance level 1% if the 99% confidence interval for the sample mean m does not include v. – Reject the H 0 with significance level 5% if the 95% confidence interval for the sample mean m does not include v. – Reject the H 0 with significance level 10% if the 90% confidence interval for the sample mean m does not include v. t test method for the test of H0 :  = v. H a :  ≠ v. – Build the t statistic (m-v)/SE – Reject the H0 with significance level 1% if the t statistic is outside the range [-t 0.005, t ] – Reject the H0 with significance level 5% if the t statistic is outside the range [-t 0.025, t ] – Reject the H0 with significance level 10% if the t statistic is outside the range [-t 0.05, t 0.05 ] I will not ask for the one sided t-test, but good to know for your future life.

Coming up: Readings: Mid term on Tuesday, November 25. – Coverage: up to Chapter 6 inclusive. No online quiz this week. Make sure you come to sessions and recitations. For help: Amine Ouazad Office 1135, Social Science building Office hour: Tuesday from 5 to 6.30pm. GAF: Irene Paneda Sunday recitations. At the Academic Resource Center, Monday from 2 to 4pm.