Deck 4: Random Variables and Probability Distributions

Question 1

A) II only
B) I only
C) IV only
D) I, II, III, and IV
E) III only

Answer 1

D

Question 2

A recent article in the paper claims that business ethics are at an all-time low. Reporting on a recent sample, the paper claims that 41% of all employees believe their company president possesses low ethical standards. Suppose 20 of a company's employees are randomly and independently sampled and asked if they believe their company president has low ethical standards and their years of experience at the company. Could the probability distribution for the number of years of experience be modelled by a binomial probability distribution?

A) Yes, the sample is a random and independent sample.
B) No, a binomial distribution requires only two possible outcomes for each experimental unit sampled.
C) $\text { Yes, the sample size is } n = 20 \text {. }$
D) No, the employees would not be considered independent in the present sample.

Answer 2

No, a binomial distribution requires only two possible outcomes for each experimental unit sampled.

Question 3

A study of college students stated that 25% of all college students have at least one tattoo. In a random sample of 80 college students, let x be the number of the students that have at least one tattoo. Find the mean and standard deviation for this binomial distribution.

A) Mean = 20, Standard Deviation $= 15$
B) Mean $= 80$ , Standard Deviation $= 3.87$
C) Mean = 20, Standard Deviation $= 3.87$
D) Mean = 80, Standard Deviation $= 15$

Answer 3

Mean = 20, Standard Deviation $= 3.87$

Question 4

Which of the following statements is not a property of the normal curve?

A) symmetric about $\mu$
B) $P ( \mu - 3 \sigma < x < \mu + 3 \sigma ) \approx .997$
C) $P ( \mu - \sigma < x < \mu + \sigma ) \approx .95$
D) mound-shaped (or bell shaped)

Question 5

The time between arrivals at an ATM machine follows an exponential distribution with minutes. Find the probability that more than 25 minutes will pass between arrivals.

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

Question 6

Which one of the following suggests that the data set is not approximately normal?

A)


B) A data set with IQR $= 752$ and $s = 574$ .
C) A data set with $68 \%$ of the measurements within $\bar { x } \pm 2 \mathrm {~s}$ .
D)

Question 7

Data has been collected and a normal probability plot for one of the variables is shown below. Based on your knowledge of normal probability plots, do you believe the variable in question is normally distributed? The data are represented by the"o" symbols in the plot.

A) Yes. The plot reveals a straight line and this indicates the variable is normally distributed.
B) Yes. The plot reveals a curve and this indicates the variable is normally distributed.
C) No. The plot does not reveal a straight line and this indicates the variable is not normally distributed.

Question 8

A recent study suggested that 70% of all eligible voters will vote in the next presidential election. Suppose 20 eligible voters were randomly selected from the population of all eligible voters. Use a binomial probability table to find the probability that more than 10 but fewer than 16 of the 20 eligible voters sampled will vote in the next presidential election.

A) 0.714
B) 0.780
C) 0.845
D) 0.649

Question 9

A) 0.01818
B) 0.25455
C) 0.21818
D) 0.50909

Question 10

A paint machine dispenses dye into paint cans to create different shades of paint. The amount of dye dispensed into a can is known to have a normal distribution with a mean of 5 milliliters (ml) and a standard deviation of 0.4 ml. Answer the following questions based on this information. Find the dye amount that represents the 9th percentile of the distribution.

A) 5.536 ml
B) 4.464 ml
C) 4.936 ml
D) 4.836 ml
E) 4.964 ml

Question 11

A) 0.1
B) 0.7
C) 0.2
D) 0.3

Question 12

For a standard normal random variable, find the probability that z exceeds the value

A) 0.0495
B) 0.9505
C) 0.5495
D) 0.4505

Question 13

A recent study suggested that 70% of all eligible voters will vote in the next presidential election. Suppose 20 eligible voters were randomly selected from the population of all eligible voters. Which of the following is necessary for this problem to be analyzed using the binomial random variable?
I. There are two outcomes possible for each of the 20 voters sampled.
II. The outcomes of the 20 voters must be considered independent of one another.
III. The probability a voter will actually vote is 0.70, the probability they won't is 0.30.

A) I, II, and III
B) I only
C) III only
D) II only

Question 14

Question 15

50 students were randomly sampled and asked questions about their exercise habits. One of the questions they were asked concerned the frequency of exercise, defined to be the number of times they exercised in a week. This variable would be characterized as which type of random variable?

A) continuous
B) discrete

Question 16

It a recent study of college students indicated that 30% of all college students had at least one tattoo. A small private college decided to randomly and independently sample 15 of their students and ask if they have a tattoo. Find the standard deviation for this binomial random variable. Round to the nearest hundredth when necessary.

A) 1.77
B) 3.15
C) 4.5
D) 10.5

Question 17

Which of the following is not a method used for determining whether data are from an approximately normal distribution?

A) Find the interquartile range, $\mathrm { IQR }$ , and standard deviation, $s$ , for the sample. Then $\frac { \mathrm { IQR } } { s } \approx 1.3$ .
B) Compute the intervals $\bar { x } \pm s , \bar { x } \pm 2 s$ , and $\bar { x } \pm 3 s$ . The percentages of measurements falling in each should be approximately $68 \% , 95 \%$ , and $100 \%$ respectively.
C) Construct a histogram or stem-and-leaf display. The shape of the graph or display should be uniform (evenly distributed).
D) Construct a normal probability plot. The points should fall approximately on a straight line.

Question 18

Question 19

A recent study suggested that 70% of all eligible voters will vote in the next presidential election. Suppose 20 eligible voters were randomly selected from the population of all eligible voters. How many of the sampled voters do we expect to vote in the next presidential election?

A) 14
B) 0.7
C) 0.3
D) 6

Question 20

The time between arrivals at an ATM machine follows an exponential distribution with minutes. Find the probability that less than 25 minutes will pass between arrivals.

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

Question 21

Question 22

Which binomial probability is represented on the screen below?

A) The probability of 2 successes in 8 trials where the probability of failure is .3.
B) The probability of 2 successes in 8 trials where the probability of success is .3.
C) The probability of 8 successes in 2 trials where the probability of success is .3.
D) The probability of 8 failures in 2 trials where the probability of failure is .3.

Question 23

A discrete random variable x can assume five possible values: 2, 3, 5, 8, 10. Its probability distribution is shown below. Find the probability that the random variable x is a value greater than 5.

$$\begin{array} { c | c c c c c } \mathrm { x } & 2 & 3 & 5 & 8 & 10 \\\hline \mathrm { p } ( \mathrm { x } ) & 0.10 & 0.20 & 0.30 & 0.30 & 0.10\end{array}$$

A) 0.70
B) 0.60
C) 0.40
D) 0.30

Question 24

For a binomial distribution, which probability is not equal to the probability of 1 success in 5 trials where the probability of success is .4?

A) the probability of 1 success in 5 trials where the probability of failure is .6
B) the probability of 4 failures in 5 trials where the probability of success is .6
C) the probability of 4 failures in 5 trials where the probability of success is .4
D) the probability of 4 failures in 5 trials where the probability of failure is .6

Question 25

A study of college students stated that 25% of all college students have at least one tattoo. In a random sample of 80 college students, let x be the number of the students that have at least one tattoo. Can the normal approximation be used to estimate the binomial distribution in this problem?

A) Yes
B) No

Question 26

Which one of the following suggests that the data set is approximately normal?

A) A data set with $Q _ { 1 } = 14 , Q _ { 3 } = 68$ , and $s = 41$ .
B) A data set with $Q _ { 1 } = 105 , Q _ { 3 } = 270$ , and $s = 33$ .
C) A data set with $Q _ { 1 } = 2.2 , Q _ { 3 } = 7.3$ , and $s = 2.1$ .
D) A data set with $Q _ { 1 } = 1330 , Q _ { 3 } = 2940$ , and $\mathrm { s } = 2440$ .

Question 27

The age of customers at a local hardware store follows a uniform distribution over the interval from 18 to 60 years old. Find the average age of customers to this hardware store.

A) 60 years old
B) 50 years old
C) 45 years old
D) 39 years old

Question 28

A) 0.297
B) 1
C) 0.545
D) 0.739

Question 29

Question 30

The price of a gallon of milk follows a normal distribution with a mean of $3.20 and a standard deviation of $0.10. Find the price for which 12.3% of milk vendors exceeded.

A) $3.316
B) $3.084
C) $3.238
D) $3.215

Question 31

A recent study suggested that 70% of all eligible voters will vote in the next presidential election. Suppose 20 eligible voters were randomly selected from the population of all eligible voters. Use a binomial probability table to find the probability that more than 12 of the eligible voters sampled will vote in the next presidential election.

A) 0.772
B) 0.228
C) 0.392
D) 0.887
E) 0.608

Question 32

The age of customers at a local hardware store follows a uniform distribution over the interval from 18 to 60 years old. Find the probability that the next customer who walks through the door exceeds 50 years old. Round to the nearest ten-thousandth.

A) 0.2381
B) 0.7619
C) 0.3600
D) 0.8333

Question 33

After a particular heavy snowstorm, the depth of snow reported in a mountain village followed a uniform distribution over the interval from 15 to 22 inches of snow. Find the probability that a randomly selected location in this village had between 17 and 18 inches of snow. Round to the nearest ten-thousandth.

A) 0.2857
B) 0.4286
C) 0.5714
D) 0.1429

Question 34

A) 0.74545
B) 0.50909
C) 0.78182
D) 0.49091

Question 35

After a particular heavy snowstorm, the depth of snow reported in a mountain village followed a uniform distribution over the interval from 15 to 22 inches of snow. Find the standard deviation of the snowfall amounts.

A) 1.42 inches
B) 2.02 inches
C) 18.5 inches
D) 4.08 inches

Question 36

A) 0.087
B) 0.182
C) 0.095
D) 0.818
E) 0.905

Question 37

A) 4
B) 12
C) 1
D) 3

Question 38

Question 39

A discrete random variable x can assume five possible values: 2, 3, 5, 8, 10. Its probability distribution is shown below. Find the standard deviation of the distribution.

A) 2.532
B) 5.7
C) 6.41
D) 1.845

Question 40

The price of a gallon of milk follows a normal distribution with a mean of $3.20 and a standard deviation of $0.10. What proportion of the milk vendors had prices that were less than $3.075 per gallon?

A) 0.8944
B) 0.2112
C) 0.3944
D) 0.1056

Question 41

It a recent study of college students indicated that 30% of all college students had at least one tattoo. A small private college decided to randomly and independently sample 15 of their students and ask if they have a tattoo. Use a binomial probability table to find the probability that exactly 5 of the students reported that they did have at least one tattoo.

A) 0.207
B) 0.722
C) 0.218
D) 0.515

Question 42

Suppose the number of babies born each hour at a hospital follows a Poisson distribution with a mean of 3. Find the probability that exactly six babies are born during a randomly selected hour.

Question 43

$\text { Compute } \left( \begin{array} { l } 6 \\2\end{array} \right) ( .3 ) ^ { 2 } ( .7 ) ^ { 6 - 2 }$

Question 44

An airline has requests for standby flights at half of the usual one-way air fare. Past experience has shown that these passengers have about a 1 in 5 chance of getting on the standby flight. When they fail to get on a flight as a standby, the only other choice is to fly first class on the next flight out. Suppose that the usual one-way air fare to a certain city is $158 and the cost of flying first class is $385. Should a passenger who wishes to fly to this city opt to fly as a standby? [Hint: Find the expected cost of the trip for a person flying
standby.]

Question 45

Consider the given discrete probability distribution. Construct a graph for p

Question 46

Suppose that 67% of the employees of a company participate in the company's medical savings program. Let x be the number of employees who participate in the program in a random sample of 50 employees. Find the mean and standard deviation of $x$

Question 47

The time between arrivals at an ATM machine follows an exponential distribution with minutes. Find the probability that between 15 and 25 minutes will pass between arrivals.

A) 0.141045
B) 0.223130
C) 0.082085
D) 0.305215

Question 48

The school newspaper surveyed 100 commuter students and asked two questions. First, students were asked how many courses they were currently enrolled in. Second, the commuter students were asked to estimate how long it took them to drive to campus. Considering these two variables, number of courses would best be considered a _________ variable and drive time would be Considered a _________ variable.

A) continuous; discrete
B) continuous; continuous
C) discrete; discrete
D) discrete; continuous

Question 49

Farmers often sell fruits and vegetables at roadside stands during the summer. One such roadside stand has a daily demand for tomatoes that is approximately normally distributed with a mean of 516 tomatoes and a standard deviation of 30 tomatoes. If there are 474 tomatoes available to be sold at the roadside stand at the beginning of a day, what is the probability that they will all be sold?

Question 50

Question 51

The tread life of a particular brand of tire is a random variable best described by a normal distribution with a mean of 60,000 miles and a standard deviation of 8200 miles. If the manufacturer guarantees the tread life of the tires for the first 50,160 miles, what proportion of the tires will need to be replaced under warranty?

Question 52

A discrete random variable x can assume five possible values: 2, 3, 5, 8, 10. Its probability distribution is shown below. Find the mean of the distribution.

A) 5.0
B) 5.5
C) 5.7
D) 5.6

Question 53

The rate of return for an investment can be described by a normal distribution with mean 28% and standard deviation 3%. What is the probability that the rate of return for the investment exceeds 34%?

Question 54

The number of homeruns hit during a major league baseball game follows a Poisson distribution with a mean of 3.2. Find the probability that a randomly selected game would have exactly 5 homeruns hit.

A) 0.105
B) 0.114
C) 0.219
D) 0.895

Question 55

If a data set is normally distributed, what is the proportion of measurements you would expect to fall within ?

A) 100%
B) 50%
C) 68%
D) 95%

Question 56

A new drug is designed to reduce a person's blood pressure. Eighteen randomly selected hypertensive patients receive the new drug. Suppose the probability that a hypertensive patient's blood pressure drops if he or she is untreated is 0.5. Then what is the probability of observing 16 or more blood pressure drops in a random sample of 18 treated patients if the new drug is in fact ineffective in reducing blood pressure? Round to six decimal places.

Question 57

A paint machine dispenses dye into paint cans to create different shades of paint. The amount of dye dispensed into a can is known to have a normal distribution with a mean of 5 milliliters (ml) and a standard deviation of 0.4 ml. Answer the following questions based on this information.
What proportion of the paint cans contain less than 5.54 ml of the dye?

A) 0.9115
B) 0.9885
C) 0.5885
D) 0.0885

Question 58

You randomly select 7 students from a class with 15 male and 20 female students. What isthe probability that you will choose exactly 4 females?

Question 59

Question 60

The hypergeometric random variable x counts the number of successes in the draw of 5 elements from a set of 10 elements containing 2 successes. List the possible values of $x$ .

Question 61

An online retailer reimburses a customer's shipping charges if the customer does not receive his order within one week. Delivery time (in days) is exponentially distributed with a mean of 3.2 days. What percentage of customers have their shipping charges reimbursed?

Question 62

An automobile insurance company estimates the following loss probabilities for the nextyear on a $25,000 sports car:
$$\begin{array} { l l } \text { Total loss: } & 0.001 \\50 \% \text { loss: } & 0.01 \\25 \% \text { loss: } & 0.05 \\10 \% \text { loss: } & 0.10 \\\text { No loss: } & 0.839\end{array}$$ Assuming the company will sell only a $500 deductible policy for this model (i.e., the owner covers the first $500 damage), how much annual premium should the company charge in order to average $565 profit per policy sold?

Question 63

Find the mean and standard deviation of the probability distribution for the random variable x, which represents the number of cars per household in a small town. $$\begin{array} { c | c } x & P ( x ) \\\hline 0 & .125 \\1 & .428 \\2 & .256 \\3 & .108 \\4 & .083\end{array}$$

Question 64

The time between equipment failures (in days) at a particular factory is exponentially distributed with a mean of 4.5 days. A machine just failed and was repaired today.
a. Find the probability that another machine will fail within the next day.
b. Find the probability that there will be no more equipment failures in the next week.

Question 65

Question 66

Calculate the mean for the discrete probability distribution shown here.

Question 67

An automobile manufacturer has determined that 30% of all gas tanks that were installed on its 2002 compact model are defective. If 13 of these cars are independently sampled, what is the probability that more than half need new gas tanks?

Question 68

Farmers often sell fruits and vegetables at roadside stands during the summer. One such roadside stand has a daily demand for tomatoes that is approximately normally distributed with a mean of 131 tomatoes and a standard deviation of 30 tomatoes. How many tomatoes must be available on any given day so that there is only a 1.5% chance that
all tomatoes will be sold?

Question 69

Suppose that the random variable x has an exponential distribution with θ = 3.
a. Find the probability that x assumes a value more than three standard deviations from µ.
b. Find the probability that x assumes a value less than one standard deviation from µ.
c. Find the probability that x assumes a value within a half standard deviation of µ.

Question 70

Question 71

Question 72

The board of examiners that administers the real estate broker's examination in a certain
state found that the mean score on the test was 513 and the standard deviation was 72. If the board wants to set the passing score so that only the best 80% of all applicants pass, what is the passing score? Assume that the scores are normally distributed.

Question 73

The number of traffic accidents that occur on a particular stretch of road during a month follows a Poisson distribution with a mean of 8.8. Find the probability that fewer than two accidents will occur on this stretch of road during a month.

Question 74

The printout below contains summary statistics of the heights of a sample of 200 adult men in the United States.

Descriptive Statistics: HT $$\begin{array} { | l | l | l | l | l | l | l | l | l } \hline \text { Variable } & \mathrm { N } & \text { Mean } & \text { StDev } & \text { Minimum } & \text { Q1 } & \text { Median } & \text { Q2 } & \\\hline \text { HT } & 200 & 70.187 & 2.716 & 62.375 & 67.875 & 69.625 & 71.500 & \\\hline\end{array}$$ Use the information in the printout to determine whether the distribution of heights is approximately normal. Explain your reasoning.

Question 75

A coin is flipped 6 times. The variable x represents the number of tails obtained. List the possible values $\text { of } x . \operatorname { Is } x$ discrete or continuous? Explain.

Question 76

You test 4 items from a lot of 15. What is the probability that you will test no defective items if the lot contains 3 defective items?

Question 77

A machine is set to pump cleanser into a process at the rate of 6 gallons per minute. Upon inspection, it is learned that the machine actually pumps cleanser at a rate described by the uniform distribution over the interval 5.5 to 6.5 gallons per minute. Find the probability that the machine pumps less than 5.75 gallons during a randomly selected minute.

Question 78

For a binomial distribution, if the probability of success is .48 on the first trial, what is the
probability of failure on the second trial?

Question 79

You are performing a study about the weight of preschoolers. A previous study found the weights to be normally distributed with a mean of 30 pounds and a standard deviation of 4 pounds. You randomly sample 30 preschool children and find their weights (in pounds) to be as follows. $$\begin{array} { l l l l l l l l l l } 25 & 25 & 26 & 26.5 & 27 & 27 & 27.5 & 28 & 28 & 28.5 \\29 & 29 & 30 & 30 & 30.5 & 31 & 31 & 32 & 32.5 & 32.5 \\33 & 33 & 34 & 34.5 & 35 & 35 & 37 & 37 & 38 & 38\end{array}$$
Draw a histogram to display the data. Is it reasonable to assume that the weights are normally distributed? Why?

Question 80

The diameters of ball bearings produced in a manufacturing process can be described using a uniform distribution over the interval 2.5 to 8.5 millimeters. What is the probability of a randomly selected ball bearing having a diameter less than 4.5 millimeters?