Deck 6: The Normal Distribution and Other Continuous Distributions

Question 1

In its standardized form, the normal distribution

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

Answer 1

has a mean of 0 and a standard deviation of 1.

Question 2

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, 75.8% of the college students will take more than how many minutes when trying to find a parking spot in the library parking lot?

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

Answer 2

2.8 minutes

Question 3

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

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

Answer 3

All the above.

Question 4

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 waiting time was found to be a random variable best approximated by an exponential distribution with a mean length of waiting time equal to 3 minutes (i.e. the mean number of calls answered in a minute is 1/3). What proportion of customers having to hold more than 4.5 minutes will hang up before placing an order?

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

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 waiting time was found to be a random variable best approximated by an exponential distribution with a mean length of waiting time equal to 3 minutes (i.e. the mean number of calls answered in a minute is 1/3). Find the waiting time at which only 10% of the customers will continue to hold.

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

Question 6

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 waiting time was found to be a random variable best approximated by an exponential distribution with a mean length of waiting time equal to 2.8 minutes (i.e. the mean number of calls answered in a minute is 1/2.8). What proportion of callers is put on hold longer than 2.8 minutes?

A) 0.3679
B) 0.50
C) 0.60810
D) 0.6321

Question 7

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

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

Question 8

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. What proportion of the plan recipients would receive payments beyond age 75?

Question 9

The value of the cumulative standardized normal distribution at Z is 0.6255. The value of Z is

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

Question 10

For some value of Z, the value of the cumulative standardized normal distribution is 0.8340. The value of Z is

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

Question 11

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 12

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. What proportion of the plan recipients die before they reach the standard retirement age of 65?

Question 13

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 waiting time was found to be a random variable best approximated by an exponential distribution with a mean length of waiting time equal to 2.8 minutes (i.e. the mean number of calls answered in a minute is 1/2.8). What is the probability that a randomly selected caller is placed on hold fewer than 7 minutes?

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

Question 14

The value of the cumulative standardized normal distribution at Z is 0.8770. The value of Z is

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

Question 15

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) 0.2255
C) 0.4938
D) 0.7745

Question 16

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 17

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 waiting time was found to be a random variable best approximated by an exponential distribution with a mean length of waiting time equal to 3 minutes (i.e. the mean number of calls answered in a minute is 1/3). What proportion of customers having to hold more than 1.5 minutes will hang up before placing an order?

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

Question 18

The value of the cumulative standardized normal distribution at 1.5X is 0.9332. The value of X is

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

Question 19

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) 0.3551
B) 0.3085
C) 0.2674
D) 0.1915

Question 20

For some value of Z, the value of the cumulative standardized normal distribution is 0.2090. The value of Z is

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

Question 21

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 between 3 and 5 pounds is ________.

Question 22

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

Question 23

Suppose students arrive at an advising office at a rate of 30 per hour. Which of the following distributions would you use to determine the probability that the next two students will arrive 30 minutes apart?

A) Normal distribution
B) Poisson distribution
C) Uniform distribution
D) Exponential distribution

Question 24

Let X represent the amount of time until the next student will arrive in the library parking lot at the university. If we know that the distribution of arrival time can be modeled using an exponential distribution with a mean of 4 minutes (i.e. the mean number of arrivals is 1/4 per minute), find the probability that it will take more than 10 minutes for the next student to arrive at the library parking lot.

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

Question 25

It was believed that the probability of being hit by lightning is the same during the course of a thunderstorm. Which of the following distributions would you use to determine the probability of being hit by a lightning during the first half of a thunderstorm?

A) Normal distribution
B) Poisson distribution
C) Uniform distribution
D) Exponential distribution

Question 26

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 27

A quality control manager at a plant that produces o-rings is concerned about whether the diameter of the o-rings that are produced is conformable to the specification. She has calculated that the average diameter of the o-rings is 4.2 centimeters. She also knows that approximately 95% of the o-rings have diameters fall between 3.2 and 5.2 centimeters and almost all of the o-rings have diameters between 2.7 and 5.7 centimeters. When modeling the diameters of the o-rings, which distribution should the scientists use?

A) Uniform distribution
B) Binomial distribution
C) Normal distribution
D) Exponential distribution

Question 28

The probability that a particular brand of smoke alarm will function properly and sound an alarm in the presence of smoke is 0.8. A batch of 100,000 such alarms was produced by independent production lines. Which of the following distributions would you use to figure out the probability that at least 90,000 of them will function properly in case of a fire?

A) Exponential distribution
B) Poisson distribution
C) Normal distribution
D) Uniform distribution

Question 29

Suppose the probability of a car accident taking place anywhere on a stretch of a 20 miles highway is the same. Which of the following distributions would you use to determine the probability that a car accident will occur somewhere between the 5-mile and 15-mile posts of the highway?

A) Normal distribution
B) Poisson distribution
C) Uniform distribution
D) Exponential distribution

Question 30

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, above what weight (in pounds)do 89.80% of the weights occur?

Question 31

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) Uniform distribution
B) Binomial distribution
C) Normal distribution
D) Exponential distribution

Question 32

The weight of a randomly selected cookie from a production line can most likely be modeled by which of the following distributions?

A) Uniform distribution
B) Poisson distribution
C) Normal distribution
D) Exponential distribution

Question 33

Suppose the light bulbs in a factory burn out at a rate of 50 bulbs per month. Which of the following distributions would you use to determine the probability that the next two light bulbs will burn out 2 days apart?

A) Hypergeometric distribution
B) Poisson distribution
C) Uniform distribution
D) Exponential distribution

Question 34

Suppose the probability of finding a bark beetles-infested pine tree is the same anywhere over a piece of 100-acre national forest land. Which of the following distributions would you use to determine the probability of finding a bark beetles-infested pine tree in a piece of 10-acre national forest land?

A) Normal distribution
B) Poisson distribution
C) Uniform distribution
D) Exponential distribution

Question 35

Suppose the probability of producing a defective light bulb from a production line is the same over an interval of 90 minutes. Which of the following distributions would you use to determine the probability that a defective light bulb will be produced in a 15 minutes interval?

A) Normal distribution
B) Poisson distribution
C) Uniform distribution
D) Exponential distribution

Question 36

Let X represent the amount of time till the next student will arrive in the library parking lot at the university. If we know that the distribution of arrival time can be modeled using an exponential distribution with a mean of 4 minutes (i.e. the mean number of arrivals is 1/4 per minute), find the probability that it will take between 2 and 12 minutes for the next student to arrive at the library parking lot.

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

Question 37

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 more than 4.4 pounds is ________.

Question 38

A wheel spinning game is played with a special wheel with 24 equal segments that determine the dollar values of a single spin. Which of the following distributions can best be used to compute the probability of winning a specific dollar value in a single spin?

A) Uniform distribution
B) Binomial distribution
C) Normal distribution
D) Exponential distribution

Question 39

The amount of juice that can be squeezed from a randomly selected orange out of a box of oranges with approximately the same size can most likely be modeled by which of the following distributions?

A) Uniform distribution
B) Poisson distribution
C) Normal distribution
D) Exponential distribution

Question 40

Suppose that past history shows that 60% of college students prefer Coca-Cola. A sample of 10,000 students is to be selected. Which of the following distributions would you use to figure out the probability that at least half of them will prefer Coca-Cola?

A) Uniform distribution
B) Poisson distribution
C) Normal distribution
D) Duplicated distribution

Question 41

The amount of tea leaves in a can from a particular production line is normally distributed with μ = 110 grams and σ = 25 grams. What is the probability that a randomly selected can will contain at least 100 grams of tea leaves?

Question 42

The probability that a standard normal variable Z is positive is ________.

Question 43

The probability that a standard normal random variable, Z, is less than 5.0 is approximately 0.

Question 44

The amount of tea leaves in a can from a particular production line is normally distributed with μ = 110 grams and σ = 25 grams. What is the probability that a randomly selected can will contain between 100 and 120 grams of tea leaves?

Question 45

The probability that a standard normal random variable, Z, is between 1.00 and 3.00 is 0.1574.

Question 46

The amount of tea leaves in a can from a particular production line is normally distributed with μ = 110 grams and σ = 25 grams. What is the probability that a randomly selected can will contain less than 100 grams of tea leaves?

Question 47

If a data set is approximately normally distributed, its normal probability plot would be S-shaped.

Question 48

The amount of tea leaves in a can from a particular production line is normally distributed with μ = 110 grams and σ = 25 grams. What is the probability that a randomly selected can will contain between 100 and 110 grams of tea leaves?

Question 49

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 above 10.95 ounces.

Question 50

A worker earns $15 per hour at a plant in China and is told that only 2.5% of all workers make a higher wage. If the wage is assumed to be normally distributed and the standard deviation of wage rates is $5 per hour, the average wage for the plant is $7.50 per hour.

Question 51

The probability that a standard normal random variable, Z, is between 1.50 and 2.10 is the same as the probability Z is between -2.10 and -1.50.

Question 52

The probability that a standard normal random variable, Z, is below 1.96 is 0.4750.

Question 53

Any set of normally distributed data can be transformed to its standardized form.

Question 54

The probability that a standard normal random variable, Z, falls between -2.00 and -0.44 is 0.6472.

Question 55

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 56

The probability that a standard normal random variable, Z, falls between -1.50 and 0.81 is 0.7242.

Question 57

The "middle spread," that is the middle 50% of the normal distribution, is equal to one standard deviation.

Question 58

A normal probability plot may be used to assess the assumption of normality for a particular set of data.

Question 59

The amount of tea leaves in a can from a particular production line is normally distributed with μ = 110 grams and σ = 25 grams. What is the probability that a randomly selected can will contain between 82 and 100 grams of tea leaves?

Question 60

Theoretically, the mean, median, and the mode are all equal for a normal distribution.

Question 61

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 lower than 55?

Question 62

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 63

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 64

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 65

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 66

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 15 and 16 seconds?

Question 67

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 68

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 69

The amount of tea leaves in a can from a particular production line is normally distributed with μ = 110 grams and σ = 25 grams. Approximately 83% of the can will have at least how many grams of tea leaves?

Question 70

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 15 seconds?

Question 71

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 72

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 73

The amount of tea leaves in a can from a particular production line is normally distributed with μ = 110 grams and σ = 25 grams. What is the probability that a randomly selected can will contain less than 100 grams or more than 120 grams of tea leaves?

Question 74

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 16 seconds?

Question 75

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 76

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 60 and 75?

Question 77

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 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 20% that the time lapsed will be shorter how many seconds?

Question 79

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 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 greater than 95?