Part III: Sample Tests, Chapter 6

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CHAPTER 6 TEST FORM A 1. Each of the following curves fails to be a normal curve. Give reasons why these curves are not normal curves. 1. (a) __________________________

(a)

m

(b) __________________________

(b)

m

2. According to the Empirical rule, for a distribution that is symmetrical and bell-shaped (in particular, for a normal distribution) approximately _______ of the data values will lie within three standard deviations on each side of the mean.

2.______________________________

3. Assuming that the heights of boys in a high school basketball tournament are normally distributed, with mean 70 in. and standard deviation 2.5 in., how many boys in a group of 40 in the tournament will be taller than 75 inches?

3. _____________________________

4. Let x be a random variable that represents the length of time it takes a student to complete Dr. Gill’s Chemistry Lab Project. From long experience, it is known that x has a normal distribution with mean µ = 3.6 hours and standard deviation σ = 0.5 hour. Convert each of the following x intervals to standard z intervals. (a) x ≥ 4.5

4. (a) __________________________

(b) 3 ≤ x ≤ 4

(b) __________________________

(c) x ≤ 2.5

(c) __________________________

Convert each of the following z intervals to raw score x intervals. (d) z ≤ −1

(d) __________________________

(e) 1 ≤ z ≤ 2

(e) __________________________

(f) z ≥ 1.5

(f)___________________________

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CHAPTER 6, FORM A, PAGE 2

5. John and Joel are salesmen in different districts. In John’s district, the long term mean sales is $17,319 each month with standard deviation $684. In Joel’s district, the long term mean sales is $21,971 each month with standard deviation $495. Assume that sales in both districts follow a normal distribution. (a) Last month John sold $19,214 whereas Joel sold $22,718 worth of merchandise. Relative to the buying habits of customers in each district, does this mean Joel is a better salesman? Explain.

5. (a) __________________________

(b) Convert Joel’s sales last month to a standard z score, and do the same for John’s sales last month. Then locate both z scores under a standard normal curve. Who do you think is the better salesman? Explain your answer.

(b) __________________________

6. The length of time to complete a door assembly on an automobile factory assembly line is normally distributed with mean µ = 6.7 minutes and standard deviation σ = 2.2 minutes. For a door selected at random, what is the probability the assembly line time will be (a) 5 minutes or less?

6. (a) __________________________

(b) 10 minutes or more?

(b) __________________________

(c) between 5 and 10 minutes?

(c) __________________________

7. From long experience, it is known that the time it takes to do an oil change and lubrication job on a vehicle has a normal distribution with mean µ = 17.8 minutes and standard deviation σ = 5.2 minutes. An auto service shop will give a free lube job to any customer who must wait beyond the guaranteed time to complete the work. If the shop does not want to give more than 1% of its customers a free lube job, how long should the guarantee be (round to the nearest minute).

7. _____________________________

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Part III: Sample Tests, Chapter 6

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CHAPTER 6, FORM A, PAGE 3

8. You are examining a quality control chart regarding the number of employees absent each shift from a large manufacturing plant. The plant is staffed so that operations are still efficient when the average number of employees absent each shift is µ = 15.7 with standard deviation σ = 3.5. For the most recent 12 shifts, the number of absent employees were Shift #

1 6

2 10

3 7

4 16

5 19

6 18

7 17

8 21

9 22

10 18

11 16

12 19

(a) Make a control chart showing the number of employees absent during the 12-day period.

(b) Are there any periods during which the number absent is out of control? Identify the out of control periods according to Type I, Type II, Type III out of control signals.

8. (a)

(b) __________________________

9. Medical treatment will cure about 87% of all people who suffer from a certain eye disorder. Suppose a large medical clinic treats 57 people with this disorder. Let r be a random variable that represents the number of people that will recover. The clinic wants a probability distribution for r. (a) Write a brief but complete description in which you explain why the normal approximation to the binomial would apply. Are the assumptions satisfied? Explain.

9. (a) __________________________

(b) Estimate P ( r ≤ 47 ) .

(b) __________________________

(c) Estimate P ( 47 ≤ r ≤ 55 ) .

(c) __________________________

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CHAPTER 6 TEST FORM B 1. Each of the following curves fails to be a normal curve. Give reasons why these curves are not normal curves. 1. (a) __________________________

(a)

m

(b) __________________________

(b)

m

2. According to the Empirical rule, for a distribution that is symmetrical and bell-shaped (in particular, for a normal distribution) approximately _______ of the data values will lie within one standard deviation on each side of the mean.

2.______________________________

3. Assuming that the weights of newborn babies at a certain hospital are normally distributed with mean 6.5 pounds and standard deviation 1.2 pounds, how many babies in a group of 80 babies from this hospital will weigh more than 8.9 pounds?

3. _____________________________

4. Let x be a random variable that represents the length of time it takes a student to write a term paper for Dr. Adam’s Sociology class. After interviewing many students, it was found that x has an approximately normal distribution with mean µ = 6.8 hours and standard deviation σ = 2.1 hours. Convert each of the following x intervals to standardized z intervals.

(a) x ≤ 7.5

4. (a) __________________________

(b) 5 ≤ x ≤ 8

(b) __________________________

(c) x ≥ 4

(c) __________________________

Convert each of the following z intervals to raw score x intervals.

(d) z ≥ −2

(d) __________________________

(e) 0 ≤ z ≤ 2

(e) __________________________

(f) z ≤ 3

(f) __________________________

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Part III: Sample Tests, Chapter 6

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CHAPTER 6, FORM B, PAGE 2

5. Operating temperatures of two models of portable electric generators follow a normal distribution. For generator I, the mean temperature is µ1 = 148°F with standard deviation σ1 = 25°F. For generator II, the mean temperature is µ2 = 143°F with standard deviation σ2 = 8°F. At peak power demand, generator I was operating at 166°F, and generator II was operating at 165°F. (a) At peak power output, both generators are operating at about the same temperature. Relative to the operating characteristics, is one a lot hotter than the other? Explain. (b) Convert the peak power temperature for each generator to standard z units. Then locate both z scores under a standard normal curve. Could one generator be near a melt down? Which one? Explain your answer.

5. (a) __________________________

(b) __________________________

6. Weights of a certain model of fully loaded gravel trucks follow a normal distribution with mean µ = 6.4 tons and standard deviation σ = 0.3 ton. What is the probability that a fully loaded truck of this model is (a) less than 6 tons?

6. (a) __________________________

(b) more than 7 tons?

(b) __________________________

(c) between 6 and 7 tons?

(c) __________________________

7. Quality control studies for Speedy Jet Computer Printers show the lifetime of the printer follows a normal distribution with mean µ = 4 years and standard deviation σ = 0.78 years. The company will replace any printer that fails during the guarantee period. How long should Speedy Jet printers be guaranteed if the company wishes to replace no more than 10% of the printers?

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7. _____________________________

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CHAPTER 6, FORM B, PAGE 3

8. A toll free computer software support service for a spreadsheet program has established target length of time for each customer help phone call. The calls are targeted to have mean duration of 12 minutes with standard deviation 3 minutes. For one help technician the most recent 10 calls had the following duration. Call # Length

1 15

2 25

3 10

4 9

5 20

6 19

7 11

8 5

9 4

10 8

(a) Make a control chart showing the number of calls.

(b) Are there any periods during which the length of calls are out of control? Identify the out of control periods according to Type I, Type II, Type III out of control signals.

8. (a)

(b) __________________________

9. Psychology 231 can be taken as a correspondence course on a Pass/Fail basis. Long experience with this course shows that about 71% of the students pass. This semester 88 students are taking Psychology 231 by correspondence. Let r be a random variable that represents the number that will pass. The Psychology Department wants a probability distribution for r. (a) Write a brief but complete description in which you explain why the normal approximation to the binomial would apply. Are the assumptions satisfied? Explain.

9. (a) __________________________

(b) Estimate P ( r ≥ 60 ) .

(b) __________________________

(c) Estimate P ( 60 ≤ r ≤ 70 ) .

(c) __________________________

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Part III: Sample Tests, Chapter 6

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CHAPTER 6 TEST FORM C Write the letter of the response that best answers each problem. 1. Which of the following curves is a normal curve?

1. __________

(b)

(a)

m

m

(c)

(d)

(e)

m

m

m

2. According to the Empirical rule, for a distribution that is symmetrical and bell-shaped (in particular, for a normal distribution) approximately _______ of the data values will lie within two standard deviations on each side of the mean. (a) 75%

(b) 95%

(c) 68%

(d) 88.9%

(e) 99.7%

3. The delivery time for a package sent within the United States is normally distributed with mean of 4 days and standard deviation of approximately 1 day. If 300 packages are being sent, how many packages will arrive in less than 3 days? (a) 8

(b) 96

(c) 102

(d) 198

2. __________

3. __________

(e) 48

4. Let x be a random variable that represents the length of time it takes a student to complete a take-home exam in Dr. Larson’s psychology class. After interviewing many students, it was found that x has an approximately normal distribution with mean µ = 5.2 hours and standard deviation σ = 1.8 hours. A. Convert the x interval x ≥ 9.7 to a standard z interval. (a) z ≤ 2.5

(b) z ≥ −2.5

(c) z ≥ 4.5

(d) z ≥ 2.5

4. A. __________

(e) z ≤ −2.5

B. Convert the z interval −1.5 ≤ z ≤ 1 to a raw score x interval. (a) 2.5 ≤ x ≤ 7

(b) 3.44 ≤ x ≤ 6.66

(c) 3.7 ≤ x ≤ 12.2

(d) −7 ≤ x ≤ 2.5

(e) –3.7 ≤ x ≤ –2.3

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B. __________

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CHAPTER 6, FORM C, PAGE 2 5. Maria and Zoe are taking Biology 105, but are in different classes. Maria’s class has an average of 78% with a standard deviation of 5% on the midterm while Zoe’s class has an average of 83% with a standard deviation of 12%. Assume that scores in both classes follow a normal distribution. A. Convert Maria’s midterm score of 84 to a standard z score. (a) 0.083

(b) 0.5

(c) 0.2

(d) 1.2

5. A. __________

(e) 6

B. Convert Zoe’s midterm score of 89 to a standard z score. (a) 1.2

(b) 0.5

(c) 6

(d) 0.917

B. __________

(e) 2.2

C. Who do you think did better relative to their class? (a) Maria

(b) Zoe

(c) They performed the same

(d) Neither

C. __________

(e) Cannot determine

6. The lifetime of a SuperTough AAA battery is normally distributed with mean µ = 28.5 hours and standard deviation σ = 5.3 hours. For a battery selected at random, what is the probability that the lifetime will be A. 25 hours or less? (a) 0.7454

6. A. __________

(b) 0.6604

(c) 0.2546

(d) 0.3396

(e) 0.9999

B. 34 hours or more? (a) 0.8485

(b) 0.1515

B. __________ (c) 1.038

(d) 0.8508

(e) 0.1492

C. between 25 hours and 34 hours? (a) 0.4038

(b) 0.5962

C. __________

(c) 0.1054

(d) 0.8946

(e) 2/736

7. Quality control studies for Dependable Dishwashers show the lifetime of a dishwasher follows a normal distribution with mean µ = 8 years and standard deviation σ = 1.2 years. The company will replace any dishwasher that fails during the guarantee period. How long should the company’s dishwashers be guaranteed if the company wishes to replace no more than 2% of the dishwashers? (a) 0.16 year

(b) 0.13 year

(c) 5.5 years

(d) 10.5 years

7. __________

(e) 2.5 years

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Part III: Sample Tests, Chapter 6

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CHAPTER 6, FORM C, PAGE 3

8. When evaluating a control chart, which of the following is not a warning signal that a random variable x is out of control?

8. __________

(a) A run of nine consecutive points on one side of the center line (the line at target value µ). (b)

One point falls beyond the 3σ level.

(c)

Two points fall beyond the 3σ level.

(d)

At least two points lie beyond the 2σ level on the same side of the center line.

(e)

At least two of three consecutive points lie beyond the 2σ level on the same side of the center line.

9. Records show that 29% of all payments to a mail-order company are submitted after the due date. Suppose 50 payments are submitted this week. Let r be a random variable that represents the number of payments that are late. Use the normal approximation to the binomial to estimate A. P(r ≥ 20) (a) 0.0307

9. A. __________ (b) 0.0594

(c) 0.9406

(d) 0.9564

(e) 0.0436

B. P(20 ≤ r ≤ 25) (a) 0.0591

B. __________ (b) 0.0585

(c) 0.0431

(d) 0.0304

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(e) 0.0298