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MAT533 MAT 533 MAT-533 MAT/533 GM533 GM 533 GM-533 GM/533 MATH533 MATH 533 MATH-533 MATH/533 Final Exam

(TCO B) JR Trucking buys tires from three suppliers: Goodyear, Michelin, and Bridgestone. Data on the last 1,000 tires that were purchased are described in the table below.

Defective | Not Defective | Total | |

Goodyear | 5 | 495 | 500 |

Michelin | 6 | 294 | 300 |

Bridgestone | 10 | 190 | 200 |

Total | 21 | 979 | 1000 |

If you choose a tire at random, then find the probability that the tire

a. was made by Michelin.

b. was made by Goodyear and was defective.

c. was not defective, given that the tire was made by Bridgestone. (Points : 18)

(TCO B)** **Midwest Airlines has had an 80% on time departure rate. A random sample of 20 flights is selected. Find the probability that

a. exactly 15 flights depart on time in the sample.

b. at least 17 flights depart on time in the sample.

c. less than 11 flights depart on time in the sample. (Points : 18)

(TCO A)** **Consider the following age data, which is the result of selecting a random sample of 20 United Airlines pilots.

47 45 45 52 58 55 58 44 42 38

45 52 48 47 51 45 52 42 37 40

a. Compute the **mean**, **median**, **mode**, and **standard deviation, Q _{1}, Q_{3}, Min, and Max** for the above sample data on age of pilots.

b. In the context of this situation, interpret the Median, Q

(TCO B) The demand for gasoline at a local service station is normally distributed with a mean of 27,009 gallons per day and a standard deviation of 4,530 gallons per day.

a. Find the probability that the demand for gasoline exceeds 22,000 gallons for a given day.

b. Find the probability that the demand for gasoline falls between 20,000 and 23,000 gallons for a given day.

c. How many gallons of gasoline should be on hand at the beginning of each day so that we can meet the demand 90% of the time (i.e., the station stands a 10% chance of running out of gasoline for that day)? (Points : 18)

** **(TCO C) A transportation company wants to estimate the average length of time goods are in transit across country. A random sample of 20 shipments yields the following results.

Sample Size = 20

Sample Mean = 4.6 days

Sample Standard Deviation = 1.5 days

a. Compute the 90% confidence interval for the population mean transit time.

b. Interpret this interval.

c. How many shipments should be sampled if we wish to generate a 99% confidence interval for the population mean transit time that is accurate to within .25 days? (Points : 18)

** **

** **(TCO C) An auditor for the U.S. Postal Service wants to examine its special Two-Day Priority mail handling to determine the proportion of parcels that actually require longer than 2 days for delivery. A randomly selected sample of 100 such parcels is found to contain seven that required longer than 2 days for delivery.

a. Compute the 90% confidence interval for the population proportion of parcels that require longer than 2 days for delivery.

b. Interpret this confidence interval.

c. How large a sample size will need to be selected if we wish to have a 90% confidence interval that is accurate to within 1%? (Points : 18)

(TCO D) An investigative reporter selects a random sample of 100 lawnmower repair shops and asks them to repair a particular brand of lawnmower. In only 36 of the cases the repair is done properly. Does the sample data provide evidence to conclude that less than 40% of all lawnmower repairs shops would repair this brand of lawnmower properly (with a = .10)? 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 less than 40% of all lawnmower repairs shops would repair this brand of lawnmower properly (with a = .10)? (Points : 24)

(TCO D) At a supermarket, the average number of register mistakes per day per clerk was 18. The owner of the supermarket purchased new cash registers in an effort to decrease the number of errors. After extensive training on the new registers, the manager took a random sample of 100 clerks on randomly selected days using the new registers and found the following results.

Sample Size = 100 clerks

Sample Mean = 17.25 mistakes per day per clerk

Sample Standard Deviation = 4.35 mistakes per day per clerk

Does the sample data provide evidence to conclude that the population mean number of register mistakes per day per clerk was less than 18 (using a = .01)? 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 this sample data provide evidence (with a = .01) that the population mean number of register mistakes per day per clerk was less than 18? (Points : 24)

** **

** **(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*ˆ1in 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)

(TCO E) A newly developed low-pressure snow tire has been tested to see how it wears under normal dry weather conditions. Twenty of these tires were tested on standard passenger cars. These cars were driven at high speeds on a dry test track for varying lengths of time. We are interested in finding the relationship between hours driven (HOURS, X1), brand of car driven (BRAND, X2, where 0=Ford and 1=General Motors), and tread wear (TREAD, Y in inches). The data is found below.

Hours | Brand | Tread |

13 | 0 | 0.1 |

25 | 0 | 0.2 |

27 | 0 | 0.2 |

46 | 0 | 0.3 |

18 | 0 | 0.1 |

31 | 0 | 0.2 |

46 | 0 | 0.3 |

57 | 0 | 0.4 |

75 | 0 | 0.5 |

87 | 0 | 0.6 |

62 | 1 | 0.4 |

105 | 1 | 0.7 |

88 | 1 | 0.6 |

63 | 1 | 0.4 |

77 | 1 | 0.5 |

109 | 1 | 0.7 |

117 | 1 | 0.8 |

35 | 1 | 0.2 |

98 | 1 | 0.6 |

121 | 1 | 0.8 |

**Correlations: Hours, Brand, Tread **

** **

Hours Brand

Brand 0.670

0.001

Tread 0.996 0.632

0.000 0.003

Cell Contents: Pearson correlation

P-Value

**Regression Analysis: Tread versus Hours, Brand **

** **

The regression equation is

Tread = – 0.00146 + 0.00686 Hours – 0.0286 Brand.

Predictor Coef SE Coef T P

Constant -0.001462 0.009608 -0.15 0.881

Hours 0.0068579 0.0001741 39.40 0.000

Brand -0.02861 0.01168 -2.45 0.026

S = 0.0193885 R-Sq = 99.3% R-Sq(adj) = 99.3%

Analysis of Variance

Source DF SS MS F P

Regression 2 0.97561 0.48780 1297.65 0.000

Residual Error 17 0.00639 0.00038

Total 19 0.98200

Predicted Values for New Observations

New Obs Fit SE Fit 95% CI 95% PI

1 0.31283 0.00895 (0.29393, 0.33172) (0.26777, 0.35789)

Values of Predictors for New Observations

New Obs Hours Brand

1 50.0 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 multiple regression **t-tests** on *β*ˆ1, *β*ˆ2 (use two tailed test with (a = .10). Interpret your results.

d. Predict the tread wear for tires from General Motors that were driven for 50 hours. Use both a point estimate and the appropriate interval estimate. (Points : 31)

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