Statistics. Mathematics and Economics Coursework.

Instructions to Candidates 
 ANSWER 5 OUT OF 5 QUESTIONS  
 YOU SHOULD ATTEMPT TO JUSTIFY YOUR ANSWERS AT ALL TIMES   
             
 Question 1 
A survey of 21 business intelligence analysts, who had been in their current positions from 10 to 20 years, revealed $90,734 as the average salary. Assume a random sample with a standard deviation of $15,000.  
  i. Find the 95% confidence interval for the population mean salary. 
(5 marks) ii. Complete the following sentence: We are 95% sure that the population mean salary is at least …………………………                                              (5 marks)        
[ 10 marks]  
 Question 2 
A co-worker of yours is very pleased, having just found an R-squared value of 100%, indicating that the regression equation has explained all of the variability in Y (“profits”) based on the X variables “revenues” and “costs.” You then amaze this person by correctly guessing the values of the regression coefficients. 
 i. Explain why the result (R-squared = 100%) is reasonable— trivial, even—in 
this case.                                                                                              (5 marks) 
ii. What are the values of the regression coefficients?                            (5 marks) 
[ 10 marks] 
   Question 3 
One might expect the price of a tent to reflect various characteristics; for example, we might expect larger tents to cost more, all else equal (because they will hold more people) and heavier tents to cost less, all else equal (because they are harder to carry and therefore less desirable). Listings from a mail-order camping supply company provided the price, weight, and area of 30 tents. Results of a multiple regression analysis to predict price are shown in Table 1.   
i. For tents of a given size (i.e., area), do heavier tents cost more or less on average than lighter tents?                                                                  (2.5 marks) ii. What number from the computer output provides the answer to part a? Interpret this number and give its measurement units. Is it significant?            (2.5 marks) 
iii. Is the result from part a consistent with expectations regarding tent pricing given at the start of the problem? Explain your answer.                              (2.5 marks) 
iv. For tents of a given weight, do larger tents cost more or less on average than smaller ones?                                                                                    (2.5 marks)  
Table 1: Multiple Regression Results for Tent Pricing 
Predictor Coefficient St 
Deviation t-ratio p 
Constant 120.33 54.82 2.19 0.037 
Weight 73.17 15.37 4.76 0.000 
Area -7.517 2.546 -2.95 0.006 
   S = 99.47             R-sq = 56.7%       R-sq(adj) = 53.5% 
[ 10 marks] 
  Question 4 
As part of a decision regarding a new product launch, you want to test whether or not a large enough percentage (10% or more) of the community would be interested in purchasing it. You will launch the product only if you find convincing evidence of such demand. A survey of 400 randomly selected people in the community finds that 13.0% are willing to try your proposed new product.  i. Why is a one-sided test appropriate here?  Identify the null and research hypotheses for a one-sided test.                                                             ( 5 marks) 
 ii. Perform the test at the 5% significance level and describe the result.  
( 5 marks) 
[ 10 marks] 
 Question 5 
Before agreeing to purchase a large order of polyethylene sheaths for a particular type of high pressure oil-filled submarine power cable, a company wants to see conclusive evidence that the true standard deviation of sheath thickness is less than .05 inches. 
 i. What hypotheses should be tested?                                                       ( 2 marks) ii. In this test, what would be a Type I error?                                           ( 4 marks) iii. In this test, what would be a Type II error?                                         ( 4 marks) 
[ 10 marks]