Deck 6: The Normal Distribution and Other Continuous Distributions

Question 1

The Tampa International Airport (TIA) has been criticized for the waiting times associated with departing flights. While the critics acknowledge that many flights have little or no waiting times, their complaints deal more specifically with the longer waits attributed to some flights. The critics are interested in showing, mathematically, exactly what the problems are. Which type of distribution would best model the waiting times of the departing flights at TIA?

A) exponential distribution
B) binomial distribution
C) normal distribution
D) uniform distribution

Answer 1

A

Question 2

Scientists in the Amazon are trying to find a cure for a deadly disease that is attacking the rain forests there. One of the variables that the scientists have been measuring involves the diameter of the trunk of the trees that have been affected by the disease. Scientists have calculated that the average diameter of the diseased trees is 42 centimeters. They also know that approximately 95% of the diameters fall between 32 and 52 centimeters and almost all of the diseased trees have diameters between 27 and 57 centimeters. When modeling the diameters of diseased trees, which distribution should the scientists use?

A) binomial distribution
B) normal distribution
C) exponential distribution
D) uniform distribution

Answer 2

B

Question 3

In its standardized form, the normal distribution

A) has an area equal to 0.5.
B) has a mean of 0 and a standard deviation of 1.
C) has a mean of 1 and a variance of 0.
D) cannot be used to approximate discrete probability distributions.

Answer 3

B

Question 4

Let X represent the amount of time it takes a student to park in the library parking lot at the university. If we know that the distribution of parking times can be modeled using an exponential distribution with a mean of 4 minutes, find the probability that it will take a randomly selected student between 2 and 12 minutes to park in the library lot.

A) 0.656318
B) 0.606531
C) 0.049787
D) 0.556744

Question 5

A catalog company that receives the majority of its orders by telephone conducted a study to determine how long customers were willing to wait on hold before ordering a product. The length of time was found to be a random variable best approximated by an exponential distribution with a mean equal to 3 minutes. What proportion of customers having to hold more than 4.5 minutes will hang up before placing an order?

A) 0.22313
B) 0.51342
C) 0.48658
D) 0.77687

Question 6

TABLE 6-7
A company has 125 personal computers. The probability that any one of them will require repair on a given day is 0.15.
Referring to Table 6-7, which of the following is one of the properties required so that the binomial distribution can be used to compute the probability that no more than 2 computers will require repair on a given day?

A) A randomly selected computer on a given day will either require a repair or will not.
B) The probability that two or more computers that will require repair in a given day approaches zero.
C) The probability that a computer that will require repair in the morning is the same as that in the afternoon.
D) The number of computers that will require repair in the morning is independent of the number of computers that will require repair in the afternoon.

Question 7

A catalog company that receives the majority of its orders by telephone conducted a study to determine how long customers were willing to wait on hold before ordering a product. The length of time was found to be a random variable best approximated by an exponential distribution with a mean equal to 3 minutes. What proportion of customers having to hold more than 1.5 minutes will hang up before placing an order?

A) 0.39347
B) 0.13534
C) 0.60653
D) 0.86466

Question 8

TABLE 6-7
A company has 125 personal computers. The probability that any one of them will require repair on a given day is 0.15.
Referring to Table 6-7, which of the following is one of the properties required so that the binomial distribution can be used to compute the probability that no more than 2 computers will require repair on a given day?

A) The number of computers that will require repair in the morning is independent of the number of computers that will require repair in the afternoon.
B) The number of personal computers the company owns on a given day is fixed.
C) The probability that two or more computers that will require repair in a given day approaches zero.
D) The probability that a computer that will require repair in the morning is the same as that in the afternoon.

Question 9

The owner of a fish market determined that the average weight for a catfish is 3.2 pounds with a standard deviation of 0.8 pound. A citation catfish should be one of the top 2% in weight. Assuming the weights of catfish are normally distributed, at what weight (in pounds) should the citation designation be established?

A) 7.36 pounds
B) 1.56 pounds
C) 5.20 pounds
D) 4.84 pounds

Question 10

If we know that the length of time it takes a college student to find a parking spot in the library parking lot follows a normal distribution with a mean of 3.5 minutes and a standard deviation of 1 minute, find the probability that a randomly selected college student will take between 2 and 4.5 minutes to find a parking spot in the library parking lot.

A) 0.0919
B) 0.7745
C) 0.4938
D) 0.2255

Question 11

For some value of Z, the probability that a standard normal variable is below Z is 0.2090. What is the value of Z?

A) - 0.81
B) - 0.31
C) 1.96
D) 0.31

Question 12

If we know that the length of time it takes a college student to find a parking spot in the library parking lot follows a normal distribution with a mean of 3.5 minutes and a standard deviation of 1 minute, find the point in the distribution in which 75.8% of the college students exceed when trying to find a parking spot in the library parking lot.

A) 2.8 minutes
B) 4.2 minutes
C) 3.4 minutes
D) 3.2 minutes

Question 13

TABLE 6-7
A company has 125 personal computers. The probability that any one of them will require repair on a given day is 0.15.
Referring to Table 6-7, which of the following is one of the properties required so that the binomial distribution can be used to compute the probability that no more than 2 computers will require repair on a given day?

A) The number of computers that will require repair in the morning is independent of the number of computers that will require repair in the afternoon.
B) The probability that a computer that will require repair in the morning is the same as that in the afternoon.
C) The probability that any one of the computers that will require repair on a given day will not affect or change the probability that any other computers that will require repair on the same day.
D) The probability that two or more computers that will require repair in a given day approaches zero.

Question 14

A catalog company that receives the majority of its orders by telephone conducted a study to determine how long customers were willing to wait on hold before ordering a product. The length of time was found to be a random variable best approximated by an exponential distribution with a mean equal to 3 minutes. Find the waiting time at which only 10% of the customers will continue to hold.

A) 2.3 minutes
B) 6.9 minutes
C) 3.3 minutes
D) 13.8 minutes

Question 15

If a particular batch of data is approximately normally distributed, we would find that approximately

A) 4 of every 5 observations would fall between ±1.28 standard deviations around the mean.
B) 2 of every 3 observations would fall between ±1 standard deviation around the mean.
C) 19 of every 20 observations would fall between ±2 standard deviations around the mean.
D) all of the above

Question 16

For some positive value of Z, the probability that a standard normal variable is between 0 and Z is 0.3770. What is the value of Z?

A) 1.16
B) 0.18
C) 1.47
D) 0.81

Question 17

Which of the following about the normal distribution is not true?

A) About 2/3 of the observations fall within ±1 standard deviation from the mean.
B) It is a discrete probability distribution.
C) Its parameters are the mean, µ, and standard deviation, a.
D) Theoretically, the mean, median, and mode are the same.

Question 18

For some positive value of X, the probability that a standard normal variable is between 0 and +2X is 0.1255. What is the value of X?

A) 0.32
B) 0.99
C) 0.40
D) 0.16

Question 19

For some positive value of Z, the probability that a standard normal variable is between 0 and Z is 0.3340. What is the value of Z?

A) 0.37
B) 0.97
C) 1.06
D) 0.07

Question 20

Let X represent the amount of time it takes a student to park in the library parking lot at the university. If we know that the distribution of parking times can be modeled using an exponential distribution with a mean of 4 minutes, find the probability that it will take a randomly selected student more than 10 minutes to park in the library lot.

A) 0.670320
B) 0.082085
C) 0.329680
D) 0.917915

Question 21

TABLE 6-5
Suppose the time interval between two consecutive defective light bulbs from a production line has a uniform distribution over an interval from 0 to 90 minutes.

-Referring to Table 6-5, the probability is 75% that the time interval between two consecutive defective light bulbs will fall between which two values that are the same distance from the mean?

Question 22

A catalog company that receives the majority of its orders by telephone conducted a study to determine how long customers were willing to wait on hold before ordering a product. The length of time was found to be a random variable best approximated by an exponential distribution with a mean equal to 2.8 minutes. What is the probability that a randomly selected caller is placed on hold fewer than 7 minutes?

A) 0.082085
B) 0.9990881
C) 0.0009119
D) 0.917915

Question 23

TABLE 6-6
The interval between consecutive hits at a web site is assumed to follow an exponential distribution with an average of 40 hits per minute.
Referring to Table 6-6, what is the probability that the next hit at the web site will occur within 10 seconds after just being hit by a visitor?

Question 24

TABLE 6-3
The number of column inches of classified advertisements appearing on Mondays in a certain daily newspaper is normally distributed with population mean 320 and population standard deviation 20 inches.
Referring to Table 6-3, a single Monday is chosen at random. State in which of the following ranges the number of column inches of classified advertisement is most likely to be:

A) 300- 320
B) 320- 340
C) 310- 330
D) 330- 350

Question 25

TABLE 6-5
Suppose the time interval between two consecutive defective light bulbs from a production line has a uniform distribution over an interval from 0 to 90 minutes.
Referring to Table 6-5, what is the probability that the time interval between two consecutive defective light bulbs will be at least 50 minutes?

Question 26

You were told that the mean score on a statistics exam is 75 with the scores normally distributed. In addition, you know the probability of a score between 55 and 60 is 4.41% and that the probability of a score greater than 90 is 6.68%. What is the probability of a score between 75 and 90?

Question 27

The amount of time necessary for assembly line workers to complete a product is a normal random variable with a mean of 15 minutes and a standard deviation of 2 minutes. The probability is_____ that a product is assembled in between 10 and 12 minutes.

Question 28

In the game Wheel of Fortune, which of the following distributions can best be used to compute the probability of winning the special vacation package in a single spin?

A) binomial distribution
B) uniform distribution
C) exponential distribution
D) normal distribution

Question 29

For some positive value of X, the probability that a standard normal variable is between 0 and +1.5X is 0.4332. What is the value of X?

A) 0.50
B) 1.00
C) 1.50
D) 0.10

Question 30

You were told that the amount of time lapsed between consecutive trades on the New York Stock Exchange followed a normal distribution with a mean of 15 seconds. You were also told that the probability that the time lapsed between two consecutive trades to fall between 16 to 17 seconds was 13%. The probability that the time lapsed between two consecutive trades would fall below 13 seconds was 7%. What is the probability that the time lapsed between two consecutive trades will be between 14 and 17 seconds?

Question 31

The amount of pyridoxine (in grams) in a multiple vitamin is normally distributed with µ = 110 grams and a = 25 grams. What is the probability that a randomly selected vitamin will contain less than 100 grams of pyridoxine?

Question 32

If we know that the length of time it takes a college student to find a parking spot in the library parking lot follows a normal distribution with a mean of 3.5 minutes and a standard deviation of 1 minute, find the probability that a randomly selected college student will find a parking spot in the library parking lot in less than 3 minutes.

A) 0.2674
B) 0.1915
C) 0.3085
D) 0.3551

Question 33

The true length of boards cut at a mill with a listed length of 10 feet is normally distributed with a mean of 123 inches and a standard deviation of 1 inch. What proportion of the boards will be over 125 inches in length?

Question 34

The amount of time necessary for assembly line workers to complete a product is a normal random variable with a mean of 15 minutes and a standard deviation of 2 minutes. So, 15% of the products require more than __________minutes for assembly.

Question 35

TABLE 6-5
Suppose the time interval between two consecutive defective light bulbs from a production line has a uniform distribution over an interval from 0 to 90 minutes.

-Referring to Table 6-5, what is the probability that the time interval between two consecutive defective light bulbs will be between 10 and 20 minutes?

Question 36

A catalog company that receives the majority of its orders by telephone conducted a study to determine how long customers were willing to wait on hold before ordering a product. The length of time was found to be a random variable best approximated by an exponential distribution with a mean equal to 2.8 minutes. What proportion of callers is put on hold longer than 2.8 minutes?

A) 0.50
B) 0.367879
C) 0.60810
D) 0.632121

Question 37

The amount of time necessary for assembly line workers to complete a product is a normal random variable with a mean of 15 minutes and a standard deviation of 2 minutes. The probability is _____ that a product is assembled in less than 20 minutes.

Question 38

The amount of time necessary for assembly line workers to complete a product is a normal random variable with a mean of 15 minutes and a standard deviation of 2 minutes. The probability is _____ that a product is assembled in more than 11 minutes.

Question 39

TABLE 6-5
Suppose the time interval between two consecutive defective light bulbs from a production line has a uniform distribution over an interval from 0 to 90 minutes.
Referring to Table 6-5, the probability is 50% that the time interval between two consecutive defective light bulbs will fall between which two values that are the same distance from the mean?

Question 40

The owner of a fish market determined that the average weight for a catfish is 3.2 pounds with a standard deviation of 0.8 pound. Assuming the weights of catfish are normally distributed, the probability that a randomly selected catfish will weigh less than 2.2 pounds is_____ .

Question 41

Given that X is a normally distributed random variable with a mean of 50 and a standard deviation of 2, find the probability that X is between 47 and 54.

Question 42

TABLE 6-7
A company has 125 personal computers. The probability that any one of them will require repair on a given day is 0.15.
Referring to Table 6-7 and assuming that the number of computers that requires repair on a given day follows a binomial distribution, compute the probability that there will be between 25 and 30 computers that require repair on a given day using a normal approximation.

Question 43

The amount of time necessary for assembly line workers to complete a product is a normal random variable with a mean of 15 minutes and a standard deviation of 2 minutes. So, 60% of the products would be assembled within____ and ____minutes (symmetrically distributed about the mean).

Question 44

TABLE 6-4
John has two jobs. For daytime work at a jewelry store he is paid $200 per month, plus a commission. His monthly commission is normally distributed with mean $600 and standard deviation $40. At night he works as a waiter, for which his monthly income is normally distributed with mean $100 and standard deviation $30. John's income levels from these two sources are independent of each other.
Referring to Table 6-4, find the mean and standard deviation of John's total income from these two jobs for a given month.

Question 45

The amount of pyridoxine (in grams) in a multiple vitamin is normally distributed with µ = 110 grams and ? = 25 grams. What is the probability that a randomly selected vitamin will contain between 100 and 110 grams of pyridoxine?

Question 46

TABLE 6-2
The city manager of a large city believes that the number of reported accidents on any weekend has a normal distribution. She takes a sample of nine weekends and determines the number of reported accidents during each. The ordered array for this data is: 15, 46, 53, 54, 55, 76, 82, 256, 407.

-Referring to Table 6-2, the second standard normal quantile is ______.

Question 47

TABLE 6-7
A company has 125 personal computers. The probability that any one of them will require repair on a given day is 0.15.
Referring to Table 6-7 and assuming that the number of computers that requires repair on a given day follows a binomial distribution, compute the probability that there will be less than 25 or more than 30 computers that require repair on a given day using a normal approximation.

Question 48

You were told that the amount of time lapsed between consecutive trades on the New York Stock Exchange followed a normal distribution with a mean of 15 seconds. You were also told that the probability that the time lapsed between two consecutive trades to fall between 16 to 17 seconds was 13%. The probability that the time lapsed between two consecutive trades would fall below 13 seconds was 7%. What is the probability that the time lapsed between two consecutive trades will be between 13 and 14 seconds?

Question 49

You were told that the mean score on a statistics exam is 75 with the scores normally distributed. In addition, you know the probability of a score between 55 and 60 is 4.41% and that the probability of a score greater than 90 is 6.68%. What is the probability of a score between 55 and 90?

Question 50

Patients arriving at an outpatient clinic follow an exponential distribution at a rate of 1.5 patients per hour. What is the probability that a randomly chosen arrival to be less than 10 minutes?

Question 51

The true length of boards cut at a mill with a listed length of 10 feet is normally distributed with a mean of 123 inches and a standard deviation of 1 inch. What proportion of the boards will be less than 124 inches?

Question 52

TABLE 6-1
The manager of a surveying company believes that the average number of phone surveys completed per hour by her employees has a normal distribution. She takes a sample of 15 days output from her employees and determines the average number of surveys per hour on these days. The ordered array for this data is: 10.0, 10.1, 10.3, 10.5, 10.7, 11.2, 11.4, 11.5, 11.7,
11.8, 11.8, 12.0, 12.2, 12.2, 12.5.

-Referring to Table 6-1, the fourth standard normal quantile is _____.

Question 53

TABLE 6-1
The manager of a surveying company believes that the average number of phone surveys completed per hour by her employees has a normal distribution. She takes a sample of 15 days output from her employees and determines the average number of surveys per hour on these days. The ordered array for this data is: 10.0, 10.1, 10.3, 10.5, 10.7, 11.2, 11.4, 11.5, 11.7,
11.8, 11.8, 12.0, 12.2, 12.2, 12.5.

-Referring to Table 6-1, the first standard normal quantile is ______ .

Question 54

A food processor packages orange juice in small jars. The weights of the filled jars are approximately normally distributed with a mean of 10.5 ounces and a standard deviation of 0.3 ounce. Find the proportion of all jars packaged by this process that have weights that fall below 10.875 ounces.

Question 55

The amount of pyridoxine (in grams) in a multiple vitamin is normally distributed with µ = 110 grams and ? = 25 grams. What is the probability that a randomly selected vitamin will contain between 100 and 120 grams of pyridoxine?

Question 56

TABLE 6-4
John has two jobs. For daytime work at a jewelry store he is paid $200 per month, plus a commission. His monthly commission is normally distributed with mean $600 and standard deviation $40. At night he works as a waiter, for which his monthly income is normally distributed with mean $100 and standard deviation $30. John's income levels from these two sources are independent of each other.
Referring to Table 6-4, the probability is 0.2 that John's total income from these two jobs in a given month is less than how much?

Question 57

Suppose Z has a standard normal distribution with a mean of 0 and standard deviation of So 27% of the possible Z values are smaller than _____.

Question 58

TABLE 6-5
Suppose the time interval between two consecutive defective light bulbs from a production line has a uniform distribution over an interval from 0 to 90 minutes.
Referring to Table 6-5, what is the probability that the time interval between two consecutive defective light bulbs will be at least 80 minutes?

Question 59

TABLE 6-7
A company has 125 personal computers. The probability that any one of them will require repair on a given day is 0.15.
Referring to Table 6-7 and assuming that the number of computers that requires repair on a given day follows a binomial distribution, compute the probability that there will be more than 25 computers that require repair on a given day using a normal approximation.

Question 60

Suppose Z has a standard normal distribution with a mean of 0 and standard deviation of The probability that Z values are larger than____ is 0.3483.

Question 61

A company that sells annuities must base the annual payout on the probability distribution of the length of life of the participants in the plan. Suppose the probability distribution of the lifetimes of the participants is approximately a normal distribution with a mean of 68 years and a standard deviation of 3.5 years. Find the age at which payments have ceased for approximately 86% of the plan participants.

Question 62

TABLE 6-5
Suppose the time interval between two consecutive defective light bulbs from a production line has a uniform distribution over an interval from 0 to 90 minutes.
Referring to Table 6-5, what is the probability that the time interval between two consecutive defective light bulbs will be exactly 10 minutes?

Question 63

Patients arriving at an outpatient clinic follow an exponential distribution with mean 15 minutes. What is the probability that a randomly chosen arrival to be more than 18 minutes?

Question 64

The amount of time necessary for assembly line workers to complete a product is a normal random variable with a mean of 15 minutes and a standard deviation of 2 minutes. The probability is_____ that a product is assembled in between 16 and 21 minutes.

Question 65

You were told that the amount of time lapsed between consecutive trades on the New York Stock Exchange followed a normal distribution with a mean of 15 seconds. You were also told that the probability that the time lapsed between two consecutive trades to fall between 16 to 17 seconds was 13%. The probability that the time lapsed between two consecutive trades would fall below 13 seconds was 7%. What is the probability that the time lapsed between two consecutive trades will be longer than 17 seconds?

Question 66

The amount of time necessary for assembly line workers to complete a product is a normal random variable with a mean of 15 minutes and a standard deviation of 2 minutes. So, 70% of the products would be assembled within ______ minutes.

Question 67

Patients arriving at an outpatient clinic follow an exponential distribution at a rate of 1 patient per hour. What is the probability that a randomly chosen arrival to be more than
2.5 hours?

Question 68

Patients arriving at an outpatient clinic follow an exponential distribution with mean 15 minutes. What is the probability that a randomly chosen arrival to be less than 15 minutes?

Question 69

You were told that the amount of time lapsed between consecutive trades on the New York Stock Exchange followed a normal distribution with a mean of 15 seconds. You were also told that the probability that the time lapsed between two consecutive trades to fall between 16 to 17 seconds was 13%. The probability that the time lapsed between two consecutive trades would fall below 13 seconds was 7%. The middle 86% of the time lapsed will fall between which two numbers?

Question 70

Patients arriving at an outpatient clinic follow an exponential distribution at a rate of 1 patient per hour. What is the probability that a randomly chosen arrival to be more than 1 hour?

Question 71

You were told that the mean score on a statistics exam is 75 with the scores normally distributed. In addition, you know the probability of a score between 55 and 60 is 4.41% and that the probability of a score greater than 90 is 6.68%. What is the probability of a score between 90 and 95?

Question 72

You were told that the amount of time lapsed between consecutive trades on the New York Stock Exchange followed a normal distribution with a mean of 15 seconds. You were also told that the probability that the time lapsed between two consecutive trades to fall between 16 to 17 seconds was 13%. The probability that the time lapsed between two consecutive trades would fall below 13 seconds was 7%. The middle 60% of the time lapsed will fall between which two numbers?

Question 73

TABLE 6-7
A company has 125 personal computers. The probability that any one of them will require repair on a given day is 0.15.
Referring to Table 6-7 and assuming that the number of computers that requires repair on a given day follows a binomial distribution, compute the probability that there will be no more than 8 computers that require repair on a given day using a normal approximation.

Question 74

Suppose Z has a standard normal distribution with a mean of 0 and standard deviation of
The probability that Z is more than -0.98 is______ .

Question 75

TABLE 6-5
Suppose the time interval between two consecutive defective light bulbs from a production line has a uniform distribution over an interval from 0 to 90 minutes.
Referring to Table 6-5, what is the standard deviation of the time interval?

Question 76

Suppose Z has a standard normal distribution with a mean of 0 and standard deviation of
So 96% of the possible Z values are between ______ and ______ (symmetrically distributed about the mean).

Question 77

The true length of boards cut at a mill with a listed length of 10 feet is normally distributed with a mean of 123 inches and a standard deviation of 1 inch. What proportion of the boards will be between 121 and 124 inches?

Question 78

You were told that the amount of time lapsed between consecutive trades on the New York Stock Exchange followed a normal distribution with a mean of 15 seconds. You were also told that the probability that the time lapsed between two consecutive trades to fall between 16 to 17 seconds was 13%. The probability that the time lapsed between two consecutive trades would fall below 13 seconds was 7%. The probability is 80% that the time lapsed will be longer than how many seconds?

Question 79

TABLE 6-4
John has two jobs. For daytime work at a jewelry store he is paid $200 per month, plus a commission. His monthly commission is normally distributed with mean $600 and standard deviation $40. At night he works as a waiter, for which his monthly income is normally distributed with mean $100 and standard deviation $30. John's income levels from these two sources are independent of each other.
Referring to Table 6-4, for a given month, what is the probability that John's income as a waiter is between $70 and $160?

Question 80

You were told that the mean score on a statistics exam is 75 with the scores normally distributed. In addition, you know the probability of a score between 55 and 60 is 4.41% and that the probability of a score greater than 90 is 6.68%. What is the probability of a score between 55 and 95?