Exam 3: Review of Statistics

Your packaging company fills various types of flour into bags.Recently there have been complaints from one chain of stores: a customer returned one opened 5 pound bag which weighed significantly less than the label indicated.You view the weight of the bag as a random variable which is normally distributed with a mean of 5 pounds,and,after studying the machine specifications,a standard deviation of 0.05 pounds. (a)You take a sample of 20 bags and weigh them.Sketch below what the average pattern of individual weights might look like.Let the horizontal axis indicate the sampled bag number (1,2,…,20).On the vertical axis,mark the expected value of the weight under the null hypothesis,and two (≈ 1.96)standard deviations above and below the expected value.Draw a line through the graph for E(Y)+ 2 ,E(Y),and E(Y)- 2 .How many of the bags in a sample of 20 will you expect to weigh either less than 4.9 pounds or more than 5.1 pounds? (b)You sample 25 bags of flour and calculate the average weight.What is the distribution of the average weight of these 25 bags? Repeating the same exercise 20 times,sketch what the distribution of the average weights would look like in a graph similar to the one you drew in (b),where you have adjusted the standard error of accordingly. (c)For each of the twenty observations in (c)a 95% confidence interval is constructed.Draw these confidence intervals,using the same graph as in (c).How many of these 20 confidence intervals would you expect to weigh 5 pounds under the null hypothesis?

You have collected weekly earnings and age data from a sub-sample of 1,744 individuals using the Current Population Survey in a given year. (a)Given the overall mean of $434.49 and a standard deviation of $294.67,construct a 99% confidence interval for average earnings in the entire population.State the meaning of this interval in words,rather than just in numbers.If you constructed a 90% confidence interval instead,would it be smaller or larger? What is the intuition? (b)When dividing your sample into people 45 years and older,and younger than 45,the information shown in the table is found. Test whether or not the difference in average earnings is statistically significant.Given your knowledge of age-earning profiles,does this result make sense?

With i.i.d.sampling each of the following is true except

A scatterplot

The correlation coefficient

Degrees of freedom

The t-statistic has the following distribution:

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.

IQ scores are normally distributed with an average of 100 and a standard deviation of 16.Some research suggests that left-handed individuals have a higher IQ score than right-handed individuals.To test this hypothesis,a researcher randomly selects 132 individuals and finds that their average IQ is 103.2 with a sample standard deviation of 14.6.Using the results from the sample,can you reject the null hypothesis that left-handed people have an IQ of 100 vs.the alternative that they have a higher IQ? What critical value should you choose if the size of the test is 5%?

When testing for differences of means,the t-statistic t = , where has

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.

A type I error is

(Advanced)Unbiasedness and small variance are desirable properties of estimators.However,you can imagine situations where a trade-off exists between the two: one estimator may be have a small bias but a much smaller variance than another,unbiased estimator.The concept of "mean square error" estimator combines the two concepts.Let be an estimator of μ.Then the mean square error (MSE)is defined as follows: MSE( )= E( - μ)2.Prove that MSE( )= bias2 + var( ).(Hint: subtract and add in E( )in E( - μ)2. )

Among all unbiased estimators that are weighted averages of Y1,... ,Yn ,is

A large p-value implies

The t-statistic is defined as follows:

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 chapter 8 CPS data set (ch8_cps.xls)into a spreadsheet program such as Excel.For the exercise,use the first 500 observations only.Using data for average hourly earnings only (ahe)and years of education (yrseduc),produce a scatterplot with earnings on the vertical axis and education level on the horizontal axis.What kind of relationship does the scatterplot suggest? Confirm your impression by adding a linear trendline.Find the correlation coefficient between the two and interpret it.

The p-value is defined as follows:

An estimate is

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. 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)?