Formulas:

Each problem is worth 12 points.

1. The noise level of a company's most popular lawn mower is 82 decibels. The noise levels of 6 prototypes of a new model have mean 77 decibels and standard deviation 4 decibels. Can we conclude with 95% certainty that the new mowers are quieter?

   H_o: m = 82
   H_a: m < 82
   Assume H_o: m = 82

   Rejection Region:
   
   

   Test Statistic:
   
   

   Reject Ho. We are 95% certain that the new mowers are quieter.

   
   

2. Fifty-nine customers who use the Discover card at a certain clothing store charge an average of $98 with standard deviation $34 and 66 customers who use Visa at the same store charge an average of $81 with standard deviation $30. Construct a 90% confidence interval for the mean difference in charge amounts.

   

   

   We are 90% certain that = 17 + 9.48

   
   

3. Sponsors of a state lottery bill claim that at least 2/3 of voters support the bill. Try to dispute their claim with a sample of 840 voters, of whom 545 favor the bill. Let a = 0.05.

   H_o: p = 0.67
   H_a: p < 0.67
   Assume H_o: p = 0.67
   

   Check for normality:

   np = (840)(0.67) = 563 > 5
   nq = (840)(0.33) = 277 > 5
   

   Rejection Region:
   

   Test Statistic:
   

   Draw no conclusion.

   
   

4. Suppose a company begins to offer its customers a cash discount for early payments, hoping to reduce the average age of accounts receivable. Previously, the average age was 57 days. Under the new system, a random sample of 40 accounts had an average age of 53 days with standard deviation 26 days. Can we conclude that the new policy has lowered the average age of accounts receivable? Let a = 0.02.

   H_o: m = 57
   H_a: m < 57
   Assume H_o: m = 57

   Rejection Region:
   

   Test Statistic:
   

   Draw no conclusion.

   
   

5. A random sample of 10 claimants in the MedCare insurance plan had an average reimbursement time of 39 days with standard deviation 6 days. Ten claimants in the Partners plan had an average reimbursement time of 28 days with standard deviation 5 days. Can we conclude with 95% certainty that MedCare reimbursement times exceed Parters times?

   Two population, use t, independent samples.

   H_o: m₁ - m₂ = 0
   H_a: m₁ - m₂ > 0
   Assume H_o: m₁ - m₂ = 0

   Rejection Region:
   

   Test Statistic:
   

   Reject H_o. We are 95% certain that Medicare reimbersement times exceed partners times.

   
   

6. Suppose a store's daily sales levels last year had mean $1000 and standard deviation $120. A sample of 26 days this year had mean $970 and standard deviation $100. Can we conclude, with a = 0.05 that this year's sales fluctuate less?

   Hypothesis test of variance with one population.

   H_o: s = 120
   H_a: s < 120
   Assume H_o: s = 120

   Rejection Region:
   

   Test Statistic:
   

   Draw no conclusion.

   
   

7. The average age of a plumbing supplier's accounts receivables last year was 51.5 days. The manager draws a sample of 75 accounts to determine if there is any change this year. Find the p-value of the hypothesis test if the sample had mean age 57.3 days with standard deviation 21.0 days.

   H_o: m = 51.5
   H_a: m < 51.5
   Assume H_o: m = 5751.5

   Test Statistic:
   

   

   p = 2(.0084) = .0168

   
   

8. A savings and loan business wants to see if two real estate appraisers provide different average appraisal values. Both appraisers are asked to appraise the same 6 homes. The results are listed in the table below, in thousands of dollars. Perform the test with a = 0.05.

   Home      Appraiser 1       Appraiser 2       
      1             90         88                
      2             95         95                
      3            101         103               
      4             88         89                
      5            106         110               
      6             78         79                
   

   Matched pairs.

   H_o: m₁ - m₂ = 0 (m = 0)
   H_a: m₁ - m₂ 0 (m 0)
   Assume H_o: m₁ - m₂ = 0 (m = 0)

   Rejection Region:
   

   Test Statistic:
   
   

   Draw no conclusion.