Deck 5: Continuous Random Variables

Question 1

Suppose x is a random variable best described by a uniform probability distribution with c = 10 and d = 50. Find P(40 ≤ x ≤ 50).

A) 0.25
B) 0.1
C) 0.35
D) 0.025

Answer 1

A

Question 2

For a continuous probability distribution, the probability that x is between a and b is the same regardless of whether or not you include the endpoints, a and b, of the interval.

Answer 2

True

Question 3

High temperatures in a certain city for the month of August follow a uniform distribution over the interval 68°F to 90°F. What is the probability that the high temperature on a day in August exceeds 73°F?

A) 0.7727
B) 0.2273
C) 0.462
D) 0.0455

Answer 3

A

Question 4

The number of children in a family can be modelled using a continuous random variable.

Question 5

Suppose x is a random variable best described by a uniform probability distribution with c = 10 and d = 90. Find P(x < 42).

A) 0.4
B) 0.32
C) 0.5
D) 0.04

Question 6

Suppose x is a random variable best described by a uniform probability distribution with c = 6 and d = 14. Find the value of a that makes the following probability statement true: P(x ≥ a) = 0.6.

A) 9.2
B) 10.8
C) 9.4
D) 4.8

Question 7

Suppose x is a random variable best described by a uniform probability distribution with c = 20 and d = 60. Find P(x > 52).

A) 0.2
B) 0.08
C) 0.8
D) 0.02

Question 8

The diameters of ball bearings produced in a manufacturing process can be described using a uniform distribution over the interval 3.5 to 5.5 millimeters. What is the probability that a randomly selected ball bearing has a diameter greater than 4.6 millimeters?

A) 0.45
B) 0.8364
C) 2
D) 0.5111

Question 9

Suppose x is a uniform random variable with c = 20 and d = 60. Find P(x > 44).

A) 0.4
B) 0.6
C) 0.1
D) 0.9

Question 10

Suppose x is a random variable best described by a uniform probability distribution with c = 3 and d = 5. Find the value of a that makes the following probability statement true: P(x ≤ a) = 0.75.

A) 4.5
B) 3.5
C) 4.7
D) 1.5

Question 11

Suppose x is a uniform random variable with c = 20 and d = 50. Find P(23 < x < 45). Round to the nearest hundredth when necessary.

A) 0.73
B) 0.27
C) 0.5
D) 1

Question 12

For any continuous probability distribution, P(x = c) = 0 for all values of c.

Question 13

Suppose x is a random variable best described by a uniform probability distribution with c = 20 and d = 60. Find P(x > 60).

A) 0
B) 1
C) 0.5
D) 0.4

Question 14

Suppose x is a random variable best described by a uniform probability distribution with c = 30 and d = 90. Find P(30 ≤ x ≤ 45).

A) 0.25
B) 0.15
C) 0.35
D) 0.025

Question 15

Suppose x is a random variable best described by a uniform probability distribution with c = 60 and d = 20. Find P(x ≥ 60).

A) 1
B) 0
C) 0.5
D) 0.4

Question 16

High temperatures in a certain city for the month of August follow a uniform distribution over the interval 69°F to 99°F. Find the temperature which is exceeded by the high temperatures on 90% of the days in August.

A) 72°F
B) 96°F
C) 79°F
D) 99°F

Question 17

Suppose x is a random variable best described by a uniform probability distribution with c = 3 and d = 9. Find the value of a that makes the following probability statement true: P(3.5 ≤ x ≤ a) = 0.5.

A) 6.5
B) 6
C) 4
D) 1.2

Question 18

The total area under a probability distribution equals 1.

Question 19

Suppose x is a random variable best described by a uniform probability distribution with c = 2 and d = 6. Find the value of a that makes the following probability statement true: P(x ≤ a) = 1.

A) a ≥ 6
B)a ≥ 2
C)a ≤ 6
D)a ≤ 2

Question 20

The diameters of ball bearings produced in a manufacturing process can be described using a uniform distribution over the interval 2.5 to 4.5 millimeters. Any ball bearing with a diameter of over 4.25 millimeters or under 2.75 millimeters is considered defective. What is the probability that a randomly selected ball bearing is defective?

A) .25
B) .50
C) .75
D) 0

Question 21

Use the standard normal distribution to find $$P ( - 2.25 < z < 0 )$$

A) .4878
B) .5122
C) .6831
D) .0122

Question 22

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) 0.8333
B) 0.3600
C) 0.2381
D) 0.7619

Question 23

Which geometric shape is used to represent areas for a uniform distribution?

A) Rectangle
B) Circle
C) Bell curve
D) Triangle 2 Find Mean, Variance, Standard Deviation

Question 24

A machine is set to pump cleanser into a process at the rate of 9 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 8.5 to 9.5 gallons per minute. Find the probability that the machine pumps less than 8.75 gallons during a randomly selected minute.

Question 25

Use the standard normal distribution to find $$P ( - 2.25 < z < 1.25 )$$

A) .8821
B) .0122
C) .4878
D) .8944

Question 26

Use the standard normal distribution to find $$P ( 0 < z < 2.25 )$$

A) .4878
B) .5122
C) .8817
D) .7888

Question 27

Suppose x is a uniform random variable with c = 40 and d = 70. Find the standard deviation of x.

Question 28

A machine is set to pump cleanser into a process at the rate of 9 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 8.5 to 11.5 gallons per minute. Find the probability that between 9.0 gallons and 10.0 gallons are pumped during a randomly selected minute.

A) 0.33
B) 0.67
C) 0
D) 1

Question 29

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

Question 30

A machine is set to pump cleanser into a process at the rate of 5 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 4.5 to 7.5 gallons per minute. Find the variance of the distribution.

A) 0.75
B) 0.33
C) 2.08
D) 12.00

Question 31

A machine is set to pump cleanser into a process at the rate of 10 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 10.0 to 13.0 gallons per minute. Would you expect the machine to pump more than 12.85 gallons per minute?

A) No, since .05 is a low probability.
B) Yes, since .95 is a high probability.
C) Yes, since .05 is a high probability.
D) No, since .95 is a high probability.

Question 32

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) 0.1429
C) 0.4286
D) 0.5714

Question 33

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

Question 34

A machine is set to pump cleanser into a process at the rate of 7 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 6.5 to 9.5 gallons per minute. What is the probability that at the time the machine is checked it is pumping more than 8.0 gallons per minute?

A) .50
B) .7692
C) .667
D) .25

Question 35

Suppose a uniform random variable can be used to describe the outcome of an experiment with outcomes ranging from 30 to 80. What is the mean outcome of this experiment?

A) 55
B) 30
C) 60
D) 80

Question 36

High temperatures in a certain city for the month of August follow a uniform distribution over the interval 75°F to 95°F. What is the probability that a random day in August has a high temperature that exceeds 80°F?

Question 37

Suppose x is a uniform random variable with c = 10 and d = 80. Find the mean of the random variable x.

Question 38

The diameters of ball bearings produced in a manufacturing process can be described using a uniform distribution over the interval 6.5 to 8.5 millimeters. What is the mean diameter of ball bearings produced in this manufacturing process?

A) 7.5 millimeters
B) 7.0 millimeters
C) 8.0 millimeters
D) 8.5 millimeters

Question 39

Suppose a uniform random variable can be used to describe the outcome of an experiment with outcomes ranging from 40 to 80. What is the probability that this experiment results in an outcome less than 50? Round to the nearest hundredth when necessary.

A) 0.25
B) 0.08
C) 0.31
D) 1

Question 40

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?

Question 41

Use the standard normal distribution to find $$\mathrm { P } ( \mathrm { z } < - 2.33 \text { or } \mathrm { z } > 2.33 )$$

A) .0198
B) .9809
C) .7888
D) .0606

Question 42

The amount of soda a dispensing machine pours into a 12-ounce can of soda follows a normal distribution with a mean of 12.54 ounces and a standard deviation of 0.36 ounce. Each can holds a maximum of 12.90 ounces of soda. Every can that has more than 12.90 ounces of soda poured into it causes a spill and the can must go through a special cleaning process before it can be sold. What is the probability that a randomly selected can will need to go through this process?

A) .1587
B) .3413
C) .8413
D) .6587

Question 43

Use the standard normal distribution to find $$P ( - 2.50 < z < 1.50 )$$

A) .9270
B) .8822
C) .6167
D) .5496

Question 44

Which shape is used to represent areas for a normal distribution?

A) Bell curve
B) Circle
C) Rectangle
D) Triangle

Question 45

The weight of corn chips dispensed into a 24-ounce bag by the dispensing machine has been identified as possessing a normal distribution with a mean of 24.5 ounces and a standard deviation of 0.2 ounce. What proportion of the 24-ounce bags contain more than the advertised 24 ounces of chips?

A) .9938
B) .4938
C) .0062
D) .5062

Question 46

The amount of soda a dispensing machine pours into a 12-ounce can of soda follows a normal distribution with a standard deviation of 0.16 ounce. Every can that has more than 12.40 ounces of soda poured into it causes a spill and the can must go through a special cleaning process before it can be sold. What is the mean amount of soda the machine should dispense if the company wants to limit the percentage that must be cleaned because of spillage to 3%?

A) 12.0992 ounces
B) 12.7008 ounces
C) 12.0528 ounces
D) 12.7472 ounces

Question 47

A physical fitness association is including the mile run in its secondary-school fitness test. The time for this event for boys in secondary school is known to possess a normal distribution with a mean of 460 seconds and a standard deviation of 40 seconds. The fitness association wants to recognize the fastest 10% of the boys with certificates of recognition. What time would the boys need to beat in order to earn a certificate of recognition from the fitness association?

A) 408.8 seconds
B) 511.2 seconds
C) 525.8 seconds
D) 394.2 seconds

Question 48

A physical fitness association is including the mile run in its secondary-school fitness test. The time for this event for boys in secondary school is known to possess a normal distribution with a mean of 460 seconds and a standard deviation of 50 seconds. Between what times do we expect approximately 95% of the boys to run the mile?

A) between 362 and 558 seconds
B) between 377.75 and 542.28 seconds
C) between 365 and 555 seconds
D) between 0 and 542.28 seconds

Question 49

Find a value of the standard normal random variable z, called z0, such that P(-z0 ≤ z ≤ z0) = 0.98.

A) 2.33
B) 1.645
C) 1.96
D) .99

Question 50

Find a value of the standard normal random variable z, called z0, such that P(z ≤ z0) = 0.70.

A) .53
B) .98
C) .81
D) .47

Question 51

The mean of the standard normal distribution is 1 and the standard deviation is 0.

Question 52

P(-1 < x < 0) = P(0 < x < 1) for any random variable x that is normally distributed.

Question 53

A physical fitness association is including the mile run in its secondary-school fitness test. The time for this event for boys in secondary school is known to possess a normal distribution with a mean of 440 seconds and a standard deviation of 50 seconds. Find the probability that a randomly selected boy in secondary school can run the mile in less than 325 seconds.

A) .0107
B) .4893
C) .9893
D) .5107

Question 54

Before a new phone system was installed, the amount a company spent on personal calls followed a normal distribution with an average of $500 per month and a standard deviation of $50 per month. Refer to such expenses as PCE's (personal call expenses). Find the point in the distribution below which 2.5% of the PCE's fell.

A) $402.00
B) $598.00
C) $12.50
D) $487.50

Question 55

Find a value of the standard normal random variable z, called z0, such that P(z ≥ z0) = 0.70.

A) -.53
B) -.98
C) -.81
D) -.47

Question 56

Before a new phone system was installed, the amount a company spent on personal calls followed a normal distribution with an average of $700 per month and a standard deviation of $50 per month. Refer to such expenses as PCE's (personal call expenses). Using the distribution above, what is the probability that during a randomly selected month PCE's were between $575.00 and $790.00?

A) .9579
B) .0421
C) .9999
D) .0001

Question 57

For a standard normal random variable, find the probability that z exceeds the value -1.65.

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

Question 58

The volume of soda a dispensing machine pours into a 12-ounce can of soda follows a normal distribution with a mean of 12.54 ounces and a standard deviation of 0.36 ounce. The company receives complaints from consumers who actually measure the amount of soda in the cans and claim that the volume is less than the advertised 12 ounces. What proportion of the soda cans contain less than the advertised 12 ounces of soda?

A) .0668
B) .5668
C) .9332
D) .4332

Question 59

For a standard normal random variable, find the point in the distribution in which 11.9% of the z -values fall below.

A) -1.18
B) 1.18
C) -0.30
D) -1.45

Question 60

Nearly 100% of the observed occurrences of a random variable x that is normally distributed will fall within three standard deviations of the mean of the distribution of x.

Question 61

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 419 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 62

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

Question 63

The rate of return for an investment can be described by a normal distribution with mean 27% and standard deviation 3%. What is the probability that the rate of return for the investment will be at least 22.5%?

Question 64

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 405 tomatoes and a standard deviation of 30 tomatoes. If there are 363 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 65

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 2600 miles. If the manufacturer guarantees the tread life of the tires for the first 56,880 miles, what proportion of the tires will need to be replaced under warranty?

Question 66

Suppose a random variable x is best described by a normal distribution with μ = 60 and σ = 8. Find the z-score that corresponds to the value x = 30.

A) -3.75
B) 3.75
C) -8D)8

Question 67

The rate of return for an investment can be described by a normal distribution with mean 47% and standard deviation 3%. What is the probability that the rate of return for the investment exceeds 53%?

Question 68

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) 0.3944
C) 0.2112
D) 0.1056

Question 69

Suppose a random variable x is best described by a normal distribution with $\mu = 60 \text { and } \sigma = 13 \text {. }$ . Find the z-score that corresponds to the value x = 112.

A) 4
B) 52
C) $\frac { 15 } { 13 }$
D) 13

Question 70

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 124 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 71

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 2900 miles. What warranty should the company use if they want 96% of the tires to outlast the warranty?

A) 54,925 miles
B) 65,075 miles
C) 57,100 miles
D) 62,900 miles

Question 72

Before a new phone system was installed, the amount a company spent on personal calls followed a normal distribution with an average of $700 per month and a standard deviation of $50 per month. Refer to such expenses as PCE's (personal call expenses). Find the probability that a randomly selected month had PCE's below $550.

A) 0.0013
B) 0.7857
C) 0.2143
D) 0.9987

Question 73

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 435 and the standard deviation was 72. If the board wants to set the passing score so that only the best 10% of all applicants pass, what is the passing score? Assume that the scores are normally distributed.

Question 74

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 1700 miles. What is the probability a certain tire of this brand will last between 56,430 miles and 56,940 miles?

A) .0180
B) .9813
C) .4920
D) .4649

Question 75

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 2300 miles. What is the probability a particular tire of this brand will last longer than 57,700 miles?

A) .8413
B) .1587
C) .2266
D) .7266

Question 76

Suppose a random variable x is best described by a normal distribution with μ = 60 and σ = 12. Find the z-score that corresponds to the value x = 60.

A) 0
B) 1
C) 12
D) 5

Question 77

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) 0.0885
B) 0.9885
C) 0.5885
D) 0.9115

Question 78

Suppose a random variable x is best described by a normal distribution with $\mu = 60 \text { and } \sigma = 5 \text {. }$ . Find the z-score that corresponds to the value x = 65.

A) 1
B) 5
C) 65
D) 12

Question 79

The preventable monthly loss at a company has a normal distribution with a mean of $4700 and a standard deviation of $40. A new policy was put into place, and the preventable loss the next month was $4460. What inference can you make about the new policy?

A) Because the probability that the monthly loss would be as low as $4460 is small, the new policy is working.
B) Because the probability that the monthly loss would be as low as $4460 is not very small, the new policy is not working.
C) While the probability that the monthly loss would be as low as $4460 is small, it is not unexpected.
D) The new policy is probably less effective than the one it replaced.

Question 80

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) $3.215
D) $3.238