Exam 3: Review of Statistics

In 2007,a study of close to 250,000 18-19 year-old Norwegian males found that first-borns have an IQ that is 2.3 points higher than those who are second-born.To see if you can find a similar evidence at your university,you collect data from 250 students,of which 140 are first-borns.After subjecting each of these individuals to an IQ test,you find that the first-borns score 108.3 with a standard deviation of 13.2,while the second borns achieve 107.1 with a standard deviation of 11.6.You hypothesize that first-borns and second-borns in a university population have identical IQs against the one-sided alternative hypothesis that first borns have higher IQs.Using a size of the test of 5%,what is your conclusion?

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

An estimator is

Math SAT scores (Y)are normally distributed with a mean of 500 and a standard deviation of 100.An evening school advertises that it can improve students' scores by roughly a third of a standard deviation,or 30 points,if they attend a course which runs over several weeks.(A similar claim is made for attending a verbal SAT course. )The statistician for a consumer protection agency suspects that the courses are not effective.She views the situation as follows: H0 : = 500 vs.H1 : = 530. (a)Sketch the two distributions under the null hypothesis and the alternative hypothesis. (b)The consumer protection agency wants to evaluate this claim by sending 50 students to attend classes.One of the students becomes sick during the course and drops out.What is the distribution of the average score of the remaining 49 students under the null,and under the alternative hypothesis? (c)Assume that after graduating from the course,the 49 participants take the SAT test and score an average of 520.Is this convincing evidence that the school has fallen short of its claim? What is the p-value for such a score under the null hypothesis? (d)What would be the critical value under the null hypothesis if the size of your test were 5%? (e)Given this critical value,what is the power of the test? What options does the statistician have for increasing the power in this situation?

The development office and the registrar have provided you with anonymous matches of starting salaries and GPAs for 108 graduating economics majors.Your sample contains a variety of jobs,from church pastor to stockbroker. (a)The average starting salary for the 108 students was $38,644.86 with a standard deviation of $7,541.40.Construct a 95% confidence interval for the starting salary of all economics majors at your university/college. (b)A similar sample for psychology majors indicates a significantly lower starting salary.Given that these students had the same number of years of education,does this indicate discrimination in the job market against psychology majors? (c)You wonder if it pays (no pun intended)to get good grades by calculating the average salary for economics majors who graduated with a cumulative GPA of B+ or better,and those who had a B or worse.The data is as shown in the accompanying table. Conduct a t-test for the hypothesis that the two starting salaries are the same in the population.Given that this data was collected in 1999,do you think that your results will hold for other years,such as 2002?

The standard error of is given by the following formula:

An estimator of the population value is unbiased if

Think of at least nine examples,three of each,that display a positive,negative,or no correlation between two economic variables.In each of the positive and negative examples,indicate whether or not you expect the correlation to be strong or weak.

The standard error for the difference in means if two random variables M and W,when the two population variances are different,is

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

An estimator of the population value is consistent if

A type II error

The accompanying table lists the height (STUDHGHT)in inches and weight (WEIGHT)in pounds of five college students.Calculate the correlation coefficient. STUDHGHT   WEIGHT 74         165 73         165 72         145 68         155 66         140

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 "CPS Data Used in Chapter 8" (ch8_cps.xls)and open it in Excel.This is a rather large data set to work with,so just copy the first 500 observations into a new Worksheet (these are rows 1 to 501). In the newly created Worksheet,mark A1 to A501,then select the Data tab and click on "sort." A dialog box will open.First select "Add level" from one of the options on the left.Then select "sort by" and choose "Northeast" and "Largest to Smallest." Repeat the same for the "South" as a second option.Finally press "ok." This should give you 209 observations for average hourly earnings for the Northeast region,followed by 205 observations for the South. a.For each of the 209 average hourly earnings observations for the Northeast region and separately for the South region,calculate the mean and sample standard deviation. b Use the appropriate test to determine whether or not average hourly earnings in the Northeast region the same as in the South region. c Find the 1%,5%,and 10% confidence interval for the differences between the two population means.Is your conclusion consistent with the test in part (b)? d In all three cases of using the confidence interval in (c),the power of the test is quite low (5%).What can you do to increase the power of the test without reducing the size of the test?

The power of the test is

When the sample size n is large,the 90% confidence interval for is

The sample covariance can be calculated in any of the following ways,with the exception of:

The net weight of a bag of flour is guaranteed to be 5 pounds with a standard deviation of 0.05 pounds.You are concerned that the actual weight is less.To test for this,you sample 25 bags.Carefully state the null and alternative hypothesis in this situation.Determine a critical value such that the size of the test does not exceed 5%.Finding the average weight of the 25 bags to be 4.7 pounds,can you reject the null hypothesis? What is the power of the test here? Why is it so low?

A low correlation coefficient implies that

When you perform hypothesis tests,you are faced with four possible outcomes described in the accompanying table. "☺" indicates a correct decision,and I and II indicate that an error has been made.In probability terms,state the mistakes that have been made in situation I and II,and relate these to the Size of the test and the Power of the test (or transformations of these).