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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 pvalue is lower than 0.05. • 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) Back to Amartya Sen: Are there really fewer women per men in India? • 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. 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 Ha: vote share > 46% ? One sided test of H0: m=v, Ha: m>=v • When testing m=v vs m>=v, use the t statistic method. – avoid the confidence interval method. Assume that m=v and build t = (m-v)/SE. ☞ Reject the null hypothesis at 95% if the t statistic is greater than +t0.05. ☞ Do not reject the null if t < 0. • Similarly for 90% and 99% (t0.10 and t0.01). Back to Cory Gardner • 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 !! 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 t0.05 for a test at 95% – For a two sided test, we use t0.025 for a test at 95%. • Ask yourself: Can I assume that the m 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 H0 : m = v. Ha: m ≠ v. CI = [ m – t * SE ; m + t * SE ] – Reject the H0 with significance level 1% if the 99% confidence interval for the sample mean m does not include v. – Reject the H0 with significance level 5% if the 95% confidence interval for the sample mean m does not include v. – Reject the H0 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 : m = v. Ha: m ≠ v. – Build the t statistic (m-v)/SE – Reject the H0 with significance level 1% if the t statistic is outside the range [-t0.005 , t0.005] – Reject the H0 with significance level 5% if the t statistic is outside the range [-t0.025 , t0.025] – Reject the H0 with significance level 10% if the t statistic is outside the range [-t0.05 , t0.05] 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 [email protected] Office hour: Tuesday from 5 to 6.30pm. • GAF: Irene Paneda [email protected] Sunday recitations. At the Academic Resource Center, Monday from 2 to 4pm.