Homework 7 STATS | Complete Solution

Please read each question carefully and answer it completely.  “Extra work” not explicitly needed can be copied into the end of document in an Appendix.  If an answer is incorrect, including work in the Appendix might allow for minimal “partial credit.”  (Note that although you are not always required to “show work” on the homework assignments, you may be required to do so on exams.)

#1.  Hypothesis Testing with Two Populations (means).  You are given sample data from two populations in the accompanying EXCEL file for this assignment.  For this question, please use α = 0.04.  Please round to two decimal places in your work/answer.
(a)  Conduct a four stage hypothesis test on whether there is a difference between the means of these two populations.  Assume populations are independent.  
(b)  Conduct a four stage hypothesis test on whether there is a difference between the means of these two populations.  Assume populations are “matched samples.”
(c)  Note that the construction of the denominator in your t-statistics is different based on the assumption of independent versus dependent populations.  Please fill in the specific denominator values into the table below:

    
“denominator”
std error

(a)
    

(b)
    

Assuming that the populations are dependent _______________________  (decreased/increased) the estimated standard error for the t-test.
(d)  Write out the formula for the variance of the difference of two normally distributed random variables.  What must be true of the COV(X,Y) so that the overall variance is lower for the difference of two dependent random variables than the variance for completely independent random variables?  (Hint: Recall the work we did with W = X – Y. To answer the second part of this question, consider the sign of the COV(X,Y).)
(e)  Use the command =covariance.s( ) in EXCEL to estimate the population covariance of these two populations.  Is this estimate consistent with what you discussed in (d)?  

#2. The “Empirical Rule” and Chebyshev Theorem are two important theories about how areas of a distribution are related to the center of that distribution.  Normal distributions must satisfy the “Empirical Rule;” the Chebyshev Theorem applies to all distributions.  For this question you will investigate whether a Chi-Square distribution also satisfies the Empirical Rule and show that it does follow Chebyshev.  

(a)  Please use a Chi-Square distribution with six degrees of freedom and fill in the following table.  Please use three decimal places in your table:

area within “k”
std dev’s of the mean
    
cdf notation     
According to the
Empirical Rule    
According to Chebyshev    
χ_6^2
 
k = 1
            
na    

k = 2
                

k = 3
                

(b)  Briefly discuss what this table demonstrates with respect to these two theories.

#3.  Consider the information for #5 on p. 492, [question begins with “John Calipari, head basketball coach … ].   
(a) Based on this information, please conduct a four-step hypothesis test to test whether the variance of basketball coach salaries is greater than 0.50. Set α = 0.05.
(b)  Would you reach a different conclusion about the hypothesis in (a) with α = 0.10?  α = 0.01?  Discuss why or why not.
(c)  Construct a 90% confidence interval for the variance of basketball coach salaries.  Based on this confidence interval, what is the 90% confidence interval for the standard deviation of salaries?  (Note that since your hypothesis test in (a) is one-tailed, there is not a direct relationship between that hypothesis test and your confidence interval.  Also, α is not the same.)

#4.  Consider the information for #21 on p. 500, [question begins with “Many smartphones, especially those …]  Please conduct a four step hypothesis to determine whether the population variance in battery hours of use is greater for talk time than internet.  Use α = 0.05.