Test Bank: Introduction to Probability and Statistics

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1. The standard deviation of any normal random variable is always equal to one. Answer: False Type: Concept Difficulty: Easy
2. For any normal random variable, the probability that the random variable will equal one is always zero. Answer: True Type: Concept Difficulty: Medium
3. The graph of a standard normal random variable is always symmetric. Answer: True Type: Concept Difficulty: Easy
4. The formula will convert any normal distribution into the “standard normal distribution. ” Answer: True Type: Concept Difficulty: Easy
5. Any normal random variable with a standard deviation equal to one is a standard normal random variable. Answer: False Type: Concept Difficulty: Medium
6. The notation X - N(4, 32) indicates a normal distribution with mean 2 and standard deviation
7. The total area under a normal curve is always equal to one. Answer: True Type: Concept Difficulty: Easy
8. The notation Z - N(0, 1) indicates a standard normal distribution. Answer: True Type: Concept Difficulty: Easy
9. The probability that a normal random variable will be within two standard deviations of its mean is approximately 0. 8. Answer: False Type: Concept Difficulty: Easy
10. The normal distribution is a continuous distribution. Answer: True Type: Concept Difficulty: Easy
11. The normal distribution can be used to approximate the binomial distribution when both np and n(1 - p) are at least five. Answer: True Type: Concept Difficulty: Easy
12. The normal distribution approximation to the binomial works best when n is large. Answer: True Type: Concept Difficulty: Easy
13. The formula can be used with both a normal and binomial distribution. Answer: True Type: Concept Difficulty: Easy Multiple Choice

Questions 4.

• Find P(-2 < Z < 2).
• A)0. 9544
• B)0. 4772
• C)0. 9772
• D)0. 6826
• E)none of the above

Answer: A Type: Computation Difficulty: Easy

Find P(-0. 5 < Z < 0. 5).

• A)0. 3830
• B)0. 1915
• C)0. 6515
• D)0. 3085
• E)None of the above

Answer: A Type: Computation Difficulty: Easy

What is the probability that a standard normal variable will be between -0. 5 and 1. 00?

• A)0. 2857
• B)0. 5328
• C)0. 6687
• D)0. 2500
• E)none of the above

Answer: B Type: Computation Difficulty: Easy

Find the probability that a standard normal random variable has a value greater than -1. 56.

• A)0. 0332
• B)0. 0594
• C)0. 9406
• D)0. 9668
• E)none of the above

Answer: C Type: Computation Difficulty: Easy

Let X be a normally distributed random variable with mean 100 and standard deviation. Find two values, a and b, symmetric about the mean, such that the probability of the random variable being between them is 0. 99.

• A)90. 5, 105. 9
• B)80. 2, 119. 8
• C)22, 78
• D)48. 5, 151. 5
• E)90. 1, 109. 9

Answer: D Type: Computation Difficulty: Medium

A professor grades his students on a normal distribution, with a mean of 75 and a standard deviation. If there are 39 students in his class, about how many scores between 80 and 90?

• A)5
• B)21
• C)8
• D)13
• E)none of the above

Answer: C Type: Computation Difficulty: Hard.

A calculator manufacturer performs a test on its calculators and finds their working life to be normally distributed, with a mean of 2,150 hours and a standard deviation of 450 hours. What should the manufacturer advertise as the life of the calculators so that 90% of the calculators are covered?

• A)2,555
• B)1,947
• C)1,410
• D)1,745
• E)1,574

Answer: E Type: Computation Difficulty: Hard

You have two stocks: A and B. The price of each stock is normally distributed. Stock A has a mean of 25 and a standard deviation of 3. Stock B also has a mean of 25, but the standard deviation is 5. If I buy stock A at $25 and sell it on a randomly chosen day in the future (without knowing its price then), what is the probability that I will make at least $2 on each share? Answer the same for stock B.

• A)0. 2514, 0. 1554
• B)0. 2514, 0. 3446
• C)0. 2486, 0. 1554
• D)0. 2486, 0. 3446
• E)none of the above

Answer: B Type: Computation Difficulty: Hard

Find two values symmetric around a mean of 20 such that they include an area equal to 0. 75. (standard deviation = 5).

• A)16. 65, 23. 35
• B)19. 25, 20. 75
• C)16. 25, 23. 75
• D)14. 25, 25. 75
• E)none of the above

Answer: D Type: Computation Difficulty: Hard

A spark plug manufacturer believes that his plug lasts an average of 30,000 miles, with a standard deviation of 2,500 miles. What is the probability that a given spark plug of this type will last 37,500 miles before replacement?

• A)0. 0228
• B)0. 0114
• C)0. 0013
• D)0. 0714
• E)0. 0833

Answer: C Type: Computation Difficulty: Medium

Fluctuations in the exchange rate of dollars against the pound sterling over a short time period were approximated by a normal distribution with a mean of 2. 01 and a standard deviation of 0. 13. What is the probability that the rate on a particular day was more than 1. 90?

• A)0. 8461
• B)0. 3023
• C)0. 8023
• D)0. 3461
• E)none of the above

Answer: C Type: Computation Difficulty: Medium

The average time it takes for a letter in the United States to reach from one place in the 48 contiguous states to another is 3. 2 days, with a standard deviation of 0. 85 days. What is the probability of a letter arriving at its destination no more than four days after mailing? Assume a normal distribution.

• A)0. 1736
• B)0. 3264
• C)0. 8264
• D)0. 6736
• E)0. 6528

Answer: C Type: Computation Difficulty: Medium

The contents of a particular bottle of shampoo marked as 150 ml are found to be 153 ml at an average, with a standard deviation of 2. 5 ml. What proportion of shampoo bottles contain less than the marked quantity? Assume a normal distribution.

• A)0. 2192
• B)0. 1151
• C)0. 4452
• D)0. 0548
• E)none of the above
• Answer: B Type: Computation Difficulty: Medium

The age of people in a town is normally distributed, with a mean of 34 years and a standard deviation of 11 years. Find two values for an age that will give an asymmetric 0. 95 probability interval.

• A)28. 78, 39. 23
• B)32. 30, 66. 30
• C)23. 55, 44. 45
• D)12. 44, 55. 56
• E)15. 91, 51. 10

Answer: D Type: Computation Difficulty: Medium

The weight of apples in a farm is normally distributed, with a mean of 110 grams, and a standard deviation of 15 grams. Find the probability that an apple selected at random will weigh between 95 and 105 grams.

• A)0. 3413
• B)0. 4706
• C)0. 1293
• D)0. 2108
• E)0. 5294

Answer: D Type: Computation Difficulty: Medium

A grocery store has a mean account receivable of $264, with a standard deviation of $55. The accounts receivable are normally distributed. What proportion of all accounts will be greater than $275?

• A)0.
• B)0. 1
• C)0. 4207
• D)0. 0793
• E)0. 0228

Answer: C Type: Computation Difficulty: Medium

A grocery store has a mean account receivable of $264, with a standard deviation of $55. The accounts receivable are approximately normally distributed. Find the value such that 45% of all the accounts exceed this value. That is, find x such that: P(X > x) = 0. 45.

• A)$257. 13
• B)$354. 48
• C)$270. 91
• D)$309. 00
• E)none of the above

Answer: C Type: Computation Difficulty: Medium 31. The waist measurement of students in a college is normally distributed. The standard deviation is known to be five inches.

It is found that 15% of the students have waist sizes less than 28 inches. What proportion of students will have waists between 30 and 35 inches?

• A)0. 3795
• B)0. 2389
• C)0. 1406
• D)0. 0983
• E)None of the above

Answer: A Type: Computation Difficulty: Hard

The IQs of the employees of a company are normally distributed, with a mean of 127 and a standard deviation of 11. What is the probability that the IQ of an employee selected at random will be between 120 and 130?

• A)0. 2389
• B)0. 3453
• C)0. 1064
• D)0. 1325
• E)0. 4638

Answer: B Type: Computation Difficulty: Medium 33.

For a normally distributed random variable with mean zero and standard deviation five, the probability that its value will be greater than -5 is:

• A)0. 4772
• B)0. 9544
• C)0. 3413
• D)0. 8413
• E)None of the above

Answer: D Type: Computation Difficulty: Medium

What is the probability that a standard normal random variable is between -0. 4 and 1. 4?

• A)0. 3413
• B)0. 4254
• C)0. 5746
• D)0. 2638
• E)None of the above

Answer: C Type: Computation Difficulty: Easy

All of the following are characteristics of the normal distribution, except:

• A) symmetric about the mean
• B) bell-shaped curve
• C) total area under the curve is always one
• D) it is a discrete distribution
• E) probability that x is equal to any specific value is zero

Answer: D Type: Computation Difficulty: Medium

Find two values symmetric about a mean of 100, the standard deviation of 10, such that they include an area equal to 0. 95.

• A)90, 110
• B)80. 4, 119. 6
• C)98. 04, 101. 96
• D)70, 130
• E)none of the above

Answer: B Type: Computation Difficulty: Medium

A tire manufacturer believes its tires will last an average of 48,000 miles, with a standard deviation of 2,000 miles. What is the probability that one of these tires, chosen at random, will last at least 50,000 miles?

• A)0. 6587
• B)0. 3413
• C)0. 1587
• D)0. 4772
• E)none of the above

Answer: C Type: Computation Difficulty: Medium

Suppose that an instructor gives an exam. This instructor wants to give those students in the top 2. 5% an A on this exam. What will the cutoff be for an A, if the average score on this exam is 80, with a standard deviation of 5?

• A)about 80
• B)about 90
• C)about 85
• D)about 86
• E)none of the above

Answer: B Type: Computation Difficulty: Hard Use the following to answer questions 51-54: LittleAir operates a fleet of regional jets on a contract basis for a major air carrier.

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