STAT 16 Assignment | Complete Solution

Name:  __________________________________________________

Please read each question carefully and answer it completely.  “Extra work” not explicitly needed can be copied into the end of document in the 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.  In the EXCEL file accompanying this assignment you are given data on 15 countries from Latin/South America and 15 countries from Europe/Central Asia for “Government final consumption as a % of GDP.”  Note that these countries are all “non-high income countries,” i.e. all these countries can be considered “developing” or “emerging.”  Data is taken from: http://data.worldbank.org/data-catalog/world-development-indicators.  Assume this data represents random samples and that population variances are not known.  Set α = 0.10 and round your work/answers to three decimal places.
     
(i)  Conduct a four step hypothesis test to test whether there is a difference in the mean of government consumption between these two populations using techniques in 10.2 (pp. 450-454).  Be sure to use Equation 10.7 on p. 451 to find the correct/appropriate degrees of freedom.
(ii)  Consider p. 454 “Notes and comments” at the bottom of the page—explain/show how assuming the same variance for the two populations will [if it does] change your test statistic, rejection criteria, p-value, and result.

#2.  Consider the following bivariate pmf and solve for the following: 

        X    
         1    2    
    1    0.250    0.167    
Y    2    0.083    0.333    
    3    0.167    0.000    
                
(a) The marginal pmfs for X and Y.  Using these individual marginal pmfs, find E(X) and E(Y).  
(b) COV (X,Y).

#3. On the last page of this assignment you will find a t distribution table taken from another textbook.  Note that unlike your textbook, most textbooks only have a one-page t table which means that exact critical values and distribution information are not available for all possible degrees of freedom.  In classes that use textbooks like this (without using EXCEL or other advanced technology) if the exact degrees of freedom are not listed then students must make a choice.  Often students will choose the “closest” degrees of freedom.  

I have always viewed data as “information” and therefore told classes I have taught that they should always “underestimate” what they know, not “overestimate” so that they should also choose the closest df lower than the true value.  In this question you will investigate the possible consequences of using a higher or lower value for the degrees of freedom for a hypothesis test.  Note that using a different df means you are actually using the WRONG distribution to hypothesis test.

Please fill in the table and then fill-in the correct responses to the follow-up questions.  For each column, be sure to use the distribution (degrees of freedom) listed for that column.  

Assume that you are conducting a two-tailed t-test with df = 37 and α = 0.05.  For this hypothesis test you have computed a t-test statistic of 2.035.

(a)
    
    (a) Use df = 30
“underestimate your df”
    
(b) Use df = 40
“closest table value”    
(c) df = 37
“true df”

Critical values
(+ and -)
            

p-value for 
your test stat

Hypothesis Test Result
            

  
(b) Using the true distribution with 37 df, find the F(-crit) for (a) and (b).  (The cdf evaluated at the negative critical value from columns (a) and (b).)  Please report four decimal places in your answers here.

(c) (i)  If the you use df = 30 then the P(Type I Error) in this test is (greater than/less than)  

____________________  you intend.

(ii)  If the you use df = 40 then the P(Type I Error) in this test is (greater than/less than)  

____________________  you intend.

(d)  Briefly discuss what the table and your responses above demonstrate about what it might mean if you do not use the appropriate distribution when you hypothesis test.

#4.  Consider the information for #35 on p. 469, [question begins with “In a test of the quality of two television commercials …”].   Based on this information, please conduct a four-step hypothesis test whether the proportion of people who see Commercial B and can recall its message is less than the proportion of people who see Commercial A and can recall its message.  Set α = 0.05.  (Hint: make sure you construct this test statistic with the pooled estimator for p.