MATH 533 FINAL EXAM (SPRING 2016)

Page 1

 

Question 1. 1. (TCO A) An insurance company researcher conducted a survey on the number of car thefts in a large city for a period of 20 days last summer. The results are as follows.
           

52              62              51              50              69
58              77              66              53              57
75              56              55              67              73
59              59              68              65              72

 
a. Compute the mean, median, mode, and standard deviation, Q1, Q3, Min, and Max for the above sample data on number of car thefts.
b. In the context of this situation, interpret the Median, Q1, and Q3. (Points : 33)

2. (TCO B) Consider the following data on newly hired employees in relation to which part of the country they were born and their highest degree attained.
 

HS

BS

MS

PHD

Total

East

3

5

2

1

11

Midwest

7

9

2

0

18

South

5

8

6

2

21

West

1

7

8

6

22

Total

16

29

18

9

72

 
If you choose one person at random, then find the probability that the person
 
a. is from the Midwest.
b. is from the South and has a PHD.
c. is from the West, given that person only holds a MS degree. (Points : 18)

3. (TCO B) A source in the Internal Revenue Service has stated that historically 90% of federal tax returns filed are free of arithmetic errors. A random sample of 25 returns are selected and checked carefully for arithmetic errors. Assuming independence, find the probability that
 
a. all 25 returns are free of arithmetic errors.
b. at most 23 returns are free of arithmetic errors.

c. more than 17 are free of arithmetic errors. (Points : 18)

4. (TCO B) At a local supermarket the monthly customer expenditure follows a normal distribution with a mean of $495 and a standard deviation of $121.
 
a. Find the probability that the monthly customer expenditure is less than $300 for a randomly selected customer.
b. Find the probability that the monthly customer expenditure is between $300 and $600 for a randomly selected customer.
c. The management of a supermarket wants to adopt a new promotional policy giving a free gift to every customer who spends more than a certain amount per month at this supermarket. Management plans to give free gifts to the top 8% of its customers (in terms of their expenditures). How much must a customer spend in a month to qualify for the free gift? (Points : 18)

5. (TCO C) DJ Car Rental wants to estimate the average number of miles traveled per day by each of its cars rented in California. A random sample of 110 cars rented in California yields the following results.
 
Sample Size = 110
Sample Mean = 85.5 mi
Sample Standard Deviation = 19.3 mi

a. Construct the 99% confidence interval for the average number of miles traveled per day by each of its cars rented in California.
b. Interpret this interval.
c. How many cars should be sampled if we wish to construct a 99% confidence interval for the average number of miles traveled per day that is accurate to within 2 mi? (Points : 18)

6. (TCO C) A marketing research firm wishes to estimate the percentage of homeowners who are dissatisfied with their present homeowner’s insurance policy. A simple random sample of 400 homeowners led to 80 who were dissatisfied with their homeowner’s insurance policy.
 
a. Compute the 95% confidence interval for the population percentage of homeowners who are dissatisfied with their present homeowner’s insurance policy.
b. Interpret this confidence interval.
c. How many homeowners should be sampled in order to be 95% confident of being within 2% of the population percentage of homeowners who are dissatisfied with their present homeowner’s insurance policy? (Points : 18)

7. (TCO D) A contract dispute between the National Football League and the Player’s Association arose regarding the retirement system. The NFL agreed to a settlement only if it could be shown convincingly that less than 60% of the players retired with 5 years or less playing time in their careers. A random sample of 200 retired NFL players is selected with 116 having played for 5 years or less. Does the sample data provide evidence to conclude that the percentage of players retiring with 5 years or less of playing time is less than 60% (using a = .01)?
 
a. Formulate the null and alternative hypotheses.
b. State the level of significance.
c. Find the critical value (or values), and clearly show the rejection and nonrejection regions.
d. Compute the test statistic.
e. Decide whether you can reject Ho and accept Ha or not.
f. Explain and interpret your conclusion in part e.  What does this mean?
g. Determine the observed p-value for the hypothesis test and interpret this value.  What does this mean?
h. Does this sample data provide evidence (with a = .01), that the percentage of players retiring with 5 years or less of playing time is less than 60%? (Points : 24)

8. (TCO D) A restaurant franchise company has a policy of opening new restaurants only in areas that have a mean household income in excess of $65,000. The company is currently considering an area to open a new restaurant. A random sample of 144 households from this area is selected yielding the following results.
 
Sample Size = 144
Sample Mean = $66,124
Sample Standard Deviation = $7,400
 
Does the sample data provide evidence to conclude that the population mean annual household income is in excess of $65,000 (using a = .05)? Use the hypothesis testing procedure outlined below.
 
a. Formulate the null and alternative hypotheses.
b. State the level of significance.
c. Find the critical value (or values), and clearly show the rejection and nonrejection regions.
d. Compute the test statistic.
e. Decide whether you can reject Ho and accept Ha or not.
f. Explain and interpret your conclusion in part e. What does this mean?
g. Determine the observed p-value for the hypothesis test and interpret this value. What does this mean?
h. Does the sample data provide evidence to conclude that the population mean annual household income is in excess of $65,000 (using a = .05)? (Points : 24)

Page 2

Question 1. 1. (TCO E) The Central Company manufactures a certain item once a week in a batch production run. The number of items produced in each run varies from week to week as demand fluctuates. The company is interested in the relationship between the size of the production run (SIZE, X) and the number of person-hours of labor (LABOR, Y). A random sample of 13 production runs is selected, yielding the data below.
 

SIZE

LABOR

PREDICT

40

83

60

30

60

100

70

138

90

180

50

97

60

118

70

140

40

75

80

159

70

140

40

75

80

159

70

144

50

90

60

125

50

87

 
 
Correlations: SIZE, LABOR
 
Pearson correlation of SIZE and LABOR = 0.990
P-Value = 0.000

 
Regression Analysis: EMP. versus FLIGHTS
 
The regression equation is
LABOR = - 6.16 + 2.07 SIZE
 
 
Predictor     Coef  SE Coef       T      P
Constant    -6.155    5.297   -1.16  0.270
SIZE       2.07371  0.08717   23.79  0.000
 
 
S = 5.20753   R-Sq = 98.1%   R-Sq(adj) = 97.9%
 
 
Analysis of Variance
 
Source          DF      SS      MS       F      P
Regression       1   15349   15349  565.99  0.000
Residual Error  11     298      27
Total           12   15647
 
 
Predicted Values for New Observations
 
New Obs      Fit  SE Fit       95% CI            95% PI
      1   118.27    1.45  (115.07, 121.46)  (106.37, 130.17)
      2   201.22    3.90  (192.64, 209.80)  (186.90, 215.53)X
 
X denotes a point that is an extreme outlier in the predictors.
 
 
Values of Predictors for New Observations
 
New Obs  SIZE
      1    60
      2   100
 
a. Analyze the above output to determine the regression equation.
b. Find and interpret β˖1in the context of this problem.
c. Find and interpret the coefficient of determination (r-squared).
d. Find and interpret coefficient of correlation.
e. Does the data provide significant evidence (a = .05) that the size of the production run can be used to predict the total labor hours? Test the utility of this model using a two-tailed test. Find the observed p-value and interpret.
f. Find the 95% confidence interval for the mean total labor hours for all occurrences of having production runs of size 60. Interpret this interval.
g. Find the 95% prediction interval for the total labor hours for one occurrence of a production run of size 60. Interpret this interval.
h. What can we say about the total labor hours when we had a production run of size 100? (Points : 48)

Page 3

1. (TCO E) At an auction, a national car rental agency sold 12 comparably equipped 3-year-old Chevrolet Corsicas. The data on mileage (X1), type of car (X2), and selling price (Y) are found below.

        Y = PRICE ($)
        X1= MILEAGE (miles)
        X2= TYPE (dummy variable 0=sedan, 1=coupe)

The data is given below (in MINITAB).

PRICE

Mileage

Type

PredMile

PredType

7000

60000

0

54000

0

8500

52000

0

54000

1

7000

62000

0

8900

48000

0

7600

55000

0

7200

60000

1

8500

50000

1

7800

53000

1

7200

58000

1

9000

48000

1

7200

60000

1

7700

55000

0

 
 
Correlations: PRICE, Mileage, Type
 
           PRICE    Mileage
Mileage   -0.970
           0.000
 
Type       0.023    -0.053
           0.942     0.871
 
 
Cell Contents: Pearson correlation
               P-Value
 
 
Regression Analysis: PRICE versus Mileage, Type
 
The regression equation is
PRICE = 15841 - 0.146 Mileage - 39 Type.
 
 
Predictor     Coef  SE Coef      T      P
Constant   15840.9    671.4   23.59  0.000
Mileage   -0.14562  0.01205  -12.09  0.000
Type         -39.5    114.1   -0.35  0.737
 
 
S = 197.278   R-Sq = 94.2%   R-Sq(adj) = 92.9%
 
 
Analysis of Variance
 
Source          DF       SS       MS      F      P
Regression       2  5689732  2844866  73.10  0.000
Residual Error   9   350268    38919
Total           11  6040000
 
 
Predicted Values for New Observations
 
New Obs     Fit   SE Fit       95% CI            95% PI
      1  7977.5     82.1  (7791.7, 8163.3)  (7494.1, 8460.9)
      2  7938.0     81.2  (7754.4, 8121.6)  (7455.4, 8420.6)
 
 
Values of Predictors for New Observations
 
New Obs   Mileage   Type
      1     54000   0.00
      2     54000   1.00
 
 
a. Analyze the above output to determine the multiple regression equation.
b. Find and interpret the multiple index of determination (R-Sq).
c. Perform the t-tests on β˖1, β˖2 (use two tailed test with (a = .05). Interpret your results.
d. Predict the price for an individual Chevy Corsica with mileage of 54,000 miles and with type being sedan. Use both a point estimate and the appropriate interval estimate. (Points : 31)