Exam 3: Review of Statistics

Your textbook states that when you test for differences in means and you assume that the two population variances are equal, then an estimator of the population variance is the following "pooled" estimator: $S _ { pooled} ^ { 2 } = \frac { 1 } { n _ { m } + n _ { w } - 2 } \left[ \sum _ { i = 1 } ^ { n _ { m } } \left( Y _ { i } - \overline { Y _ { m } } \right) ^ { 2 } + \sum _ { i = 1 } ^ { n _ { w } } \left( Y _ { i } - \bar { Y } _ { w } \right) ^ { 2 } \right]$ Explain why this pooled estimator can be looked at as the weighted average of the two variances.

Assume that you have 125 observations on the height (H)and weight (W)of your peers in college. Let ${ } ^ { S } H W$ = 68, ${ } ^ { S } H$ = 3.5, $\text { SW }$ = 29. The sample correlation coefficient is

An estimator $\hat { \mu }_y$ of the population value $\mu_y$ is unbiased if

A low correlation coefficient implies that

At the Stock and Watson (http://www.pearsonhighered.com/stock_watson)website go to Student Resources and select the option "Datasets for Replicating Empirical Results." Then select the "Test Score data set used in Chapters 4-9" (caschool.xls)and open the Excel data set. Next produce a scatterplot of the average reading score (horizontal axis)and the average mathematics score (vertical axis). What does the scatterplot suggest? Calculate the correlation coefficient between the two series and give an interpretation.

An estimate is

Among all unbiased estimators that are weighted averages of Y₁,..., Y_n $\bar { Y }$ , is

(Requires calculus.)Let Y be a Bernoulli random variable with success probability Pr(Y = 1)= p. It can be shown that the variance of the success probability p is $\frac { p ( 1 - p ) } { n }$ Use calculus to show that this variance is maximized for p = 0.5.

The size of the test

U.S. News and World Report ranks colleges and universities annually. You randomly sample 100 of the national universities and liberal arts colleges from the year 2000 issue. The average cost, which includes tuition, fees, and room and board, is $23,571.49 with a standard deviation of $7,015.52. (a)Based on this sample, construct a 95% confidence interval of the average cost of attending a university/college in the United States. (b)Cost varies by quite a bit. One of the reasons may be that some universities/colleges have a better reputation than others. U.S. News and World Reports tries to measure this factor by asking university presidents and chief academic officers about the reputation of institutions. The ranking is from 1 ("marginal")to 5 ("distinguished"). You decide to split the sample according to whether the academic institution has a reputation of greater than 3.5 or not. For comparison, in 2000, Caltech had a reputation ranking of 4.7, Smith College had 4.5, and Auburn University had 3.1. This gives you the statistics shown in the accompanying table. Reputation Category Average Cost Standard deviation of Cost Ranking >3.5 \ 29,311.31 \ 5,649.21 29 Ranking \leq3.5 \ 21,227.06 \ 6,133.38 71 Test the hypothesis that the average cost for all universities/colleges is the same independent of the reputation. What alternative hypothesis did you use? (c)What other factors should you consider before making a decision based on the data in (b)?

When testing for differences of means, the t-statistic t = $\frac { \bar { Y } m - \bar { Y } w } { S E ( \bar { Y } m - \overline { \bar { Y } w } ) }$ , where $S E \left[ \bar { Y } _ { m } - \bar { Y } _ { W } \right) = \sqrt { \frac { s _ { m } ^ { 2 } } { n _ { m } } + \frac { s _ { w } ^ { 2 } } { n _ { w } } }$ has

An estimator $\hat { \mu }_y$ of the population value $\mu_y$ is consistent if

A large p-value implies

An estimator $\hat { \mu }_Y$ of the population value $\mu_Y$ is more efficient when compared to another estimator $\bar { \mu }_Y$ , if

The following statement about the sample correlation coefficient is true.

Consider two estimators: one which is biased and has a smaller variance, the other which is unbiased and has a larger variance. Sketch the sampling distributions and the location of the population parameter for this situation. Discuss conditions under which you may prefer to use the first estimator over the second one.

The critical value of a two-sided t-test computed from a large sample

A scatterplot

The power of the test

Imagine that you had sampled 1,000,000 females and 1,000,000 males to test whether or not females have a higher IQ than males. IQs are normally distributed with a mean of 100 and a standard deviation of 16. You are excited to find that females have an average IQ of 101 in your sample, while males have an IQ of 99. Does this difference seem important? Do you really need to carry out a t-test for differences in means to determine whether or not this difference is statistically significant? What does this result tell you about testing hypotheses when sample sizes are very large?