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# Decision Science Chapter 11 Probability and Statistics

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16. A metropolitan school system consists of two districts, east and west. The east district contains 35% of all students, and the west district contains the other 65%. A vocational aptitude test was given to all students; 10% of the east district students failed, and 25% of the west district students failed. Given that a student failed the test, what is the posterior probability that the student came from the east district?

30. The manager of the local National Video Store sells videocassette recorders at discount prices. If the store does not have a video recorder in stock when a customer wants to buy one, it will lose the sale because the customer will purchase a recorder from one of the many local competitors. The problem is that the cost of renting warehouse space to keep enough recorders in inventory to meet all demand is excessively high. The manager has determined that if 90% of customer demand for recorders can be met, then the combined cost of lost sales and inventory will be minimized. The manager has estimated that monthly demand for recorders is normally distributed, with a mean of 180 recorders and a standard deviation of 60. Determine the number of recorders the manager should order each month to meet 90% of customer demand.

#### Solution Summary

The solution discusses decision science in probability and statistics.

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## Applied statistics in business and economics

Prepare answers to the following assignments from the e-text, Applied Statistics in Business and Economics, by Doane and Seward:

Chapter 10 - Chapter Exercises
10.30
In Dallas, some fire trucks were painted yellow (instead of red) to heighten their visibility. During a test period, the fleet of red fire trucks made 153,348 runs and had 20 accidents, while the fleet of yellow fire trucks made 135,035 runs and had 4 accidents. At _ = .01, did the yellow fire trucks have a significantly lower accident rate? (a) State the hypotheses. (b) State the decision rule and sketch it. (c) Find the sample proportions and z test statistic. (d) Make a decision. (e) Find the p-value and interpret it. (f) If statistically significant, do you think the difference is large enough to be important? If so, to whom, and why? (g) Is the normality assumption fulfilled? Explain.
Accident Rate for Dallas Fire Trucks
Statistic Red Fire Trucks Yellow Fire Trucks
Number of accidents x1 = 20 accidents x2 = 4 accidents
Number of fire runs n1 = 153,348 runs n2 = 135,035 runs

10.44
Does lovastatin (a cholesterol-lowering drug) reduce the risk of heart attack? In a Texas study, researchers gave lovastatin to 2,325 people and an inactive substitute to 2,081 people (average age 58). After 5 years, 57 of the lovastatin group had suffered a heart attack, compared with 97 for the inactive pill. (a) State the appropriate hypotheses. (b) Obtain a test statistic and p-value. Interpret the results at _ = .01. (c) Is normality assured? (d) Is the difference large enough to be important? (e) What else would medical researchers need to know before prescribing this drug widely? (Data are from Science News 153 [May 30, 1998], p. 343.)

10.46
To test the hypothesis that students who finish an exam first get better grades, Professor Hardtack kept track of the order in which papers were handed in. The first 25 papers showed a mean score of 77.1 with a standard deviation of 19.6, while the last 24 papers handed in showed a mean score of 69.3 with a standard deviation of 24.9. Is this a significant difference at _ = .05? (a) State the hypotheses for a right-tailed test. (b) Obtain a test statistic and p-value assuming equal variances. Interpret these results. (c) Is the difference in mean scores large enough to be important? (d) Is it reasonable to assume equal variances? (e) Carry out a formal test for equal variances at _ = .05, showing all steps clearly.

10.56
A sample of 25 concession stand purchases at the October 22 matinee of Bride of Chucky showed a mean purchase of \$5.29 with a standard deviation of \$3.02. For the October 26 evening showing of the same movie, for a sample of 25 purchases the mean was \$5.12 with a standard deviation of \$2.14. The means appear to be very close, but not the variances. At _ = .05, is there a difference in variances? Show all steps clearly, including an illustration of the decision rule. (Data are from a project by statistics students Kim Dyer, Amy Pease, and Lyndsey Smith.)

Chapter 11 - Chapter Exercise 11.24
In a bumper test, three types of autos were deliberately crashed into a barrier at 5 mph, and the resulting damage (in dollars) was estimated. Five test vehicles of each type were crashed, with the results shown below. Research question: Are the mean crash damages the same for these three vehicles? Crash1
Crash Damage (\$)
Goliath Varmint Weasel
1,600 1,290 1,090
760 1,400 2,100
880 1,390 1,830
1,950 1,850 1,250
1,220 950 1,920

Chapter 15 - Chapter Exercises
15.18
Sixty-four students in an introductory college economics class were asked how many credits they had earned in college, and how certain they were about their choice of major. Research question: At _ = .01, is the degree of certainty independent of credits earned? Certainty
Credits Earned Very Uncertain Somewhat Certain Very Certain Row Total
0-9 12 8 3 23
10-59 8 4 10 22
60 or more 1 7 11 19
Col Total 21 19 24 64

15.22
A student team examined parked cars in four different suburban shopping malls. One hundred vehicles were examined in each location. Research question: At _ = .05, does vehicle type vary by mall location? (Data are from a project by MBA students Steve Bennett, Alicia Morais, Steve Olson, and Greg Corda.) Vehicles
Vehicle Type Somerset Oakland Great Lakes Jamestown Row Total
Car 44 49 36 64 193
Minivan 21 15 18 13 67
Full-sized Van 2 3 3 2 10
SUV 19 27 26 12 84
Truck 14 6 17 9 46
Col Total 100 100 100 100 400

15.24
High levels of cockpit noise in an aircraft can damage the hearing of pilots who are exposed to this hazard for many hours. A Boeing 727 co-pilot collected 61 noise observations using a handheld sound meter. Noise level is defined as "Low" (under 88 decibels), "Medium" (88 to 91 decibels), or "High" (92 decibels or more). There are three flight phases (Climb, Cruise, Descent). Research question: At _ = .05, is the cockpit noise level independent of flight phase? (Data are from Capt. Robert E. Hartl, retired.) Noise
Noise Level Climb Cruise Descent Row Total
Low 6 2 6 14
Medium 18 3 8 29
High 1 3 14 18
Col Total 25 8 28 61

15.28
Can people really identify their favorite brand of cola? Volunteers tasted Coca-Cola Classic, Pepsi, Diet Coke, and Diet Pepsi, with the results shown below. Research question: At _ = .05, is the correctness of the prediction different for the two types of cola drinkers? Could you identify your favorite brand in this kind of test? Since it is a 2 _ 2 table, try also a two-tailed two-sample z test for ð1 = ð2 (see Chapter 10) and verify that z2 is the same as your chi-square statistic. Which test do you prefer? Why? (Data are from Consumer Reports 56, no. 8 [August 1991], p. 519.) Cola
Correct? Regular Cola Diet Cola Row Total
Yes, got it right 7 7 14
No, got it wrong 12 20 32
Col Total 19 27 46

Chapter 9 - Chapter Exercises

9.54
Faced with rising fax costs, a firm issued a guideline that transmissions of 10 pages or more should be sent by 2-day mail instead. Exceptions are allowed, but they want the average to be 10 or below. The firm examined 35 randomly chosen fax transmissions during the next year, yielding a sample mean of 14.44 with a standard deviation of 4.45 pages. (a) At the .01 level of significance, is the true mean greater than 10? (b) Use Excel to find the right-tail p-value.
Hypotheses: H0: _ = 10 vs. _ > 10
Level of Significance: _ = 0.01
Decision Rule: Reject the null hypothesis if p-value < 0.01
Calculation: Hypothesis Test: Mean vs. Hypothesized Value
10.000 hypothesized value
14.400 mean Sample
4.450 std. dev.
0.752 std. error
35 n

5.85 z
2.46E-09 p-value (one-tailed, upper)
Conclusion: Since the p-value < 0.01, the null hypothesis is rejected. Therefore, there is sufficient evidence to conclude at 0.01 level of significance that the true mean is greater than 10.

9.56
A coin was flipped 60 times and came up heads 38 times. (a) At the .10 level of significance, is the coin biased toward heads? Show your decision rule and calculations. (b) Calculate a p-value and interpret it.
Hypotheses: H0: p = 0.5 vs. HA: p > 0.5
Level of Significance: _ = 10%
Decision Rule: Reject the null hypothesis if p-value < 0.1
Calculations:
Hypothesis test for proportion vs hypothesized value
Observed Hypothesized
0.633 0.5 p (as decimal)
38/60 30/60 p (as fraction)
37.98 30. X
60 60 n

0.0645 std. error
2.06 z
.0197 p-value (one-tailed, upper)

Conclusion: Since the p-value is less than 0.1, the null hypothesis is rejected. Therefore, there is sufficient evidence to conclude at 10% level of significance that the coin biased towards heads.

9.62
The Web-based company Oh Baby! Gifts has a goal of processing 95 percent of its orders on the same day they are received. If 485 out of the next 500 orders are processed on the same day, would this prove that they are exceeding their goal, using _ = .025?
Hypotheses: H0: p = 0.95 vs. HA: p > 0.95
Level of Significance: _ = 2.5%
Decision Rule: Reject the null hypothesis if p-value < 0.025
Calculations:
Hypothesis test for proportion vs hypothesized value
Observed Hypothesized
0.97 0.95 p (as decimal)
485/500 475/500 p (as fraction)
485. 475. X
500 500 N

0.0097 std. error
2.05 Z
.0201 p-value (one-tailed, upper)

Conclusion: Since the p-value < 0.025, the null hypothesis is rejected. Therefore, there is sufficient evidence to conclude at 2.5% level of significance that the company is exceeding its goal of 95% being processed on the same day.

9.64
An auditor reviewed 25 oral surgery insurance claims from a particular surgical office, determining that the mean out-of-pocket patient billing above the reimbursed amount was \$275.66 with a standard deviation of \$78.11. (a) At the 5 percent level of significance, does this sample prove a violation of the guideline that the average patient should pay no more than \$250 out-of-pocket? State your hypotheses and decision rule. (b) Is this a close decision?

Hypotheses: H0: _ = 250 vs. _ > 250
Level of Significance: _ = 5%
Decision Rule: Reject the null hypothesis if p-value < 0.05
Calculation: Hypothesis Test: Mean vs. Hypothesized Value
250.0000 hypothesized value
275.6600 mean Sample
78.1100 std. dev.
15.6220 std. error
25 n
24 df

1.64 t
.0568 p-value (one-tailed, upper)
Conclusion: Since the p-value is less than 0.05, the null hypothesis is not rejected. Therefore, there is insufficient evidence to conclude at 5% level of significance that this sample prove a violation of the guideline that the average patient should pay no more than \$250 out of pocket.

(b) Yes, this is a very close decision because the p-value is very close to 0.05.

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