Deck 10: Probability and Calculus

Question 1

Find the value of the probability of the standard normal variable Z. Round your answer to four decimal places. ​ ​

A) 0.8522
B) ​0.4728
C) ​0.1013
D) ​0.2266
E) ​0.3717

Answer 1

​0.2266

Question 2

The scores on an economics examination are normally distributed with a mean of 62 and a standard deviation of 18. If the instructor assigns a grade of A to 10% of the class, what is the lowest score a student may have and still get an A?
​
Round your answer to the nearest integer.
​
__________ %

Answer 2

86

Question 3

Find the value of the probability of the standard normal variable Z corresponding to the shaded area under the standard normal curve. Round your answer to four decimal places.
​
​ ​
​ ​ __________

Answer 3

0.8448

Question 4

According to data released by a city's Chamber of Commerce, the weekly wages (in dollars) of female factory workers are normally distributed with a mean of 470 and a standard deviation of 40. Find the probability that a female factory worker selected at random from the city has a weekly wage of $450 to $550. Round your answer to four decimal places. ​

A) 0.103
B) 0.2224
C) 0.3842
D) 0.6687
E) 0.5754

Question 5

Find the value of the probability of the standard normal variable Z corresponding to the shaded area under the standard normal curve. Round your answer to four decimal places. ​ ​

A)
B) ​
C) ​
D) ​
E) ​

Question 6

Suppose X is a normal random variable with and . Find . Round your answer to four decimal places. ​

A)
B)
C)
D)
E)

Question 7

Find the value of the probability of the standard normal variable Z corresponding to the shaded area under the standard normal curve. Round your answer to four decimal places. ​ ​ ​

A) ​0.0516
B) ​0.1056
C) ​0.1488
D) ​0.7968
E) ​0.2640

Question 8

According to data released by a city's Chamber of Commerce, the weekly wages (in dollars) of female factory workers are normally distributed with a mean of 470 and a standard deviation of 50. Find the probability that a female factory worker selected at random from the city has a weekly wage of $450 to $550.
​
Round your answer to four decimal places.

Question 9

Let Z be the standard normal variable. Round your answers to the nearest hundredth, if necessary.
​
Find the value of z if z satisfies .
​ __________
​
Find the value of z if z satisfies .
​ __________

Question 10

Find the value of the probability of the standard normal variable Z. ​ ​

A) ​0.1912
B) ​0.7707
C) ​0.9115
D) ​0.6948
E) ​0.1616

Question 11

Find the value of the probability of the standard normal variable Z corresponding to the shaded area under the standard normal curve. Round your answer to four decimal places.
​
​ ​ ​ = __________

Question 12

Find the value of the probability of the standard normal variable Z corresponding to the shaded area under the standard normal curve. Round your answer to four decimal places.
​
​ ​ ​

Question 13

To be eligible for further consideration, applicants for certain Civil Service positions must first pass a written qualifying examination on which a score of 80 or more must be obtained. In a recent examination, it was found that the scores were normally distributed with a mean of 70 points and a standard deviation of 10 points. Determine the percentage of applicants who passed the written qualifying examination.
​
Round your answer to the nearest hundredth, if necessary.
​
__________%

Question 14

Let Z be the standard normal variable. Find the value of z if z satisfies . Round your answer to the nearest hundredth. ​

A)
B)
C)
D)
E)

Question 15

Let Z be the standard normal variable. Find the values of z if z satisfies . Round your answer to the nearest hundredth. ​

A) ​
B) ​
C) ​
D) ​
E) ​

Question 16

Find the value of the probability of the standard normal variable Z corresponding to the shaded area under the standard normal curve. Round your answer to four decimal places.
​ ​ __________

Question 17

The medical records of infants delivered at Kaiser Memorial Hospital show that the infants' lengths at birth (in inches) are normally distributed with a mean of 20 and a standard deviation of 2.7. Find the probability that an infant selected at random from among those delivered at the hospital measures more than 21 in. Round your answer to four decimal places. ​

A)
B)
C)
D)
E)

Question 18

Find the value of the probability of the standard normal variable Z corresponding to the shaded area under the standard normal curve. Round your answer to four decimal places.
​ ​ = __________

Question 19

Find the value of the probability of the standard normal variable Z corresponding to the shaded area under the standard normal curve. Round your answer to four decimal places.
​ ​ = __________

Question 20

Find the value of the probability of the standard normal variable Z corresponding to the shaded area under the standard normal curve. Round your answer to four decimal places. ​

A) 0.1638
B) ​0.2299
C) ​0.8181
D) ​0.2152
E) ​0.8295

Question 21

The life span of a certain plant species (in days) is described by the probability density function
​ ​
If a plant of this species is selected at random, how long can the plant be expected to live?
​
__________ days

Question 22

Find the median of the random variable x with the probability density function defined on the indicated interval I. The median of x is defined to be the number m such that . Observe that half of the x-values lie below m and the other half lie above m. ​ ​

A) ​
B) ​
C) ​
D) ​
E)

Question 23

The amount of snowfall in feet in a remote region of Alaska in the month of January is a continuous random variable with probability density function ​ ​
Find the amount of snowfall one can expect in any given month of January in Alaska.
​

A) 4.5 ft
B) 2.5 ft
C) 9.5 ft
D) 0.5 ft
E) 7.5 ft

Question 24

The medical records of infants delivered at Kaiser Memorial Hospital show that the infants' lengths at birth (in inches) are normally distributed with a mean of 20 and a standard deviation of 2.6. Round your answers to four decimal places, if necessary.
​
Find the probability that the length of an infant selected at random from among those delivered at the hospital is more than 22 in.
​
__________
​
Find the probability that the length of an infant selected at random from among those delivered at the hospital is less than 18 in.
​
__________
​
Find the probability that the length of an infant selected at random from among those delivered at the hospital is between 19 and 21 in.
​
__________

Question 25

The amount of time t (in minutes) a shopper spends browsing in the magazine section of a supermarket is a continuous random variable with probability density function ​
​ ​
How much time is a shopper chosen at random expected to spend in the magazine section?
​

A) minutes
B) minutes
C) minutes
D) minutes
E) minutes

Question 26

Find the median of the random variable x with the probability density function defined on the indicated interval I. The median of x is defined to be the number m such that . Observe that half of the x-values lie below m and the other half lie above m. ​ ​

A) ​
B) ​
C)
D) ​
E) ​

Question 27

The IQs of students at an Elementary School were measured and found to be normally distributed with a mean of 100 and a standard deviation of 20. Round your answers to four decimal places, if necessary.
​
Find the probability that a student selected at random will have an IQ of 140 or higher.
​
__________
​
Find the probability that a student selected at random will have an IQ of 120 or higher.
​
__________
​
Find the probability that a student selected at random will have an IQ of between 100 and 120.
​
__________
​
Find the probability that a student selected at random will have an IQ of 90 or less.
​
__________

Question 28

Find the mean of the random variable x associated with the probability density function over the indicated interval. ​ ; ​

A) ​
B) ​
C) ​
D) ​
E)

Question 29

Suppose X is a normal random variable with and . Round your answers to four decimal places, if necessary.
​
Find .
​ __________
​
Find .
​ __________
​
Find .
​ __________

Question 30

Find the mean of the random variable x associated with the probability density function over the indicated interval. ​ ; ​

A) ​
B) ​
C)
D) ​
E) ​

Question 31

Find the mean of the random variable x associated with the probability density function over the indicated interval. ​ ; ​

A) ​
B) ​
C) ​
D) ​
E)

Question 32

The quantity demanded x (in thousands of pounds) of a certain brand of butter per week is a continuous random variable with probability density function
​ ​
What is the expected demand for this brand of butter per week?
​
__________ lb/wk

Question 33

Find the mean of the random variable x associated with the probability density function over the indicated interval. ​ ; ​

A)
B) ​
C) ​
D) ​
E) ​

Question 34

Find the mean of the random variable x associated with the probability density function over the indicated interval. Round your answer to four decimal places. ​ ; ​

A) ​
B) ​
C) ​
D) ​
E) ​

Question 35

Let Z be the standard normal variable. Round your answers to the nearest hundredth, if necessary.
​
Find the value of z if z satisfies .
​ __________
​
Find the value of z if z satisfies .
​ __________

Question 36

The tread lives of Super Titan radial tires under normal driving conditions are normally distributed with a mean of 42,000 mi and a standard deviation of 5,000 mi. What is the probability that a tire selected at random will have a tread life of more than 35,000 mi? Round your answer to four decimal places, if necessary.
​
__________
​
If four new tires are installed on a car and they experience even wear, determine the probability that all four tires still have useful tread lives after 35,000 mi of driving. Round your answer to four decimal places, if necessary.
​
__________

Question 37

According to the data released by a city's Chamber of Commerce, the weekly wages of factory workers are normally distributed with a mean of $380 and a standard deviation of $50. Round your answers to four decimal places, if necessary.
​
Find the probability that a worker selected at random from the city has a weekly wage of less than $300.
​
__________
​
Find the probability that a worker selected at random from the city has a weekly wage of more than $460.
​
__________
​
Find the probability that a worker selected at random from the city has a weekly wage of between $350 and $450.
​
__________

Question 38

Find the mean of the random variable x associated with the probability density function over the indicated interval. ​ ; ​

A) ​
B) ​
C)
D) ​
E) ​

Question 39

The amount of snowfall in feet in a remote region of Alaska in the month of January is a continuous random variable with probability density function
​ ​
Find the amount of snowfall one can expect in any given month of January in Alaska. Give the answer to one decimal place, if necessary.
​
__________ ft

Question 40

Let Z be the standard normal variable. Round your answers to the nearest tenth, if necessary.
​
Find the value of z if z satisfies .
​ __________
​
Find the value of z if z satisfies .
​ __________

Question 41

Find the median of the random variable x with the probability density function defined on the indicated interval I. The median of x is defined to be the number m such that . Observe that half of the x-values lie below m and the other half lie above m. Round the answer to two decimal places, if necessary.
​ ​

Question 42

Find the median of the random variable x with the probability density function defined on the indicated interval I. The median of x is defined to be the number m such that . Observe that half of the x-values lie below m and the other half lie above m.
​ ​ __________

Question 43

Find the median of the random variable x with the probability density function defined on the indicated interval I. The median of x is defined to be the number m such that . Observe that half of the x-values lie below m and the other half lie above m. Round the answer to two decimal places, if necessary.
​ ;

Question 44

A study conducted by Uni-Mart, a mail-order department store, reveals that the time intervals in minutes between incoming telephone calls on its toll-free 600 line between 10 A.M. and 3 P.M. are exponentially distributed with probability density function .
​
What is the probability that the time interval between successive calls is more than 5 min? Give the answer to four decimal places, if necessary.

Question 45

Find the value of the constant ​k so that the function is a probability density function in the indicated interval.
​

Question 46

The amount of time t (in minutes) a shopper spends browsing in the magazine section of a supermarket is a continuous random variable with probability density function
​ ​
How much time is a shopper chosen at random expected to spend in the magazine section?

Question 47

Suppose the time intervals in seconds between arrivals of successive cars at an expressway tollbooth during rush hour are exponentially distributed with associated probability density function . Find the probability that the average time interval between arrivals of successive cars is more than 9 sec. Round your answer to four decimal places. ​

A) 0.1929
B) ​0.5564
C) ​0.9403
D) ​0.2366
E) ​0.0498

Question 48

Find the median of the random variable x with the probability density function defined on the indicated interval I. The median of x is defined to be the number m such that . Observe that half of the x-values lie below m and the other half lie above m.
​ ​ __________

Question 49

A study conducted by Uni-Mart, a mail-order department store, reveals that the time intervals in minutes between incoming telephone calls on its toll-free 600 line between 9 A.M. and 2 P.M. are exponentially distributed with probability density function . What is the probability that the time interval between successive calls is more than 6 min? Round your answer to four decimal places. ​

A) 0.4899
B) ​0.8869
C) ​0.2950
D) ​0.1737
E) ​0.8981

Question 50

The function f is a probability density function defined on the given interval. Find . ​ ​

A) ​
B) ​
C) ​
D) ​
E) ​

Question 51

Find the value of the constant k so that the function is a probability density function in the indicated interval.
​

Question 52

Find the value of the constant k so that the function is a probability density function in the indicated interval. ​ ​

A)
B) ​
C) ​
D) ​
E) ​

Question 53

Find the value of the constant k so that the function is a probability density function in the indicated interval. ​ ​

A)
B) ​
C) ​
D) ​
E) ​

Question 54

Suppose the time intervals in seconds between arrivals of successive cars at an expressway tollbooth during rush hour are exponentially distributed with associated probability density function .
​
Find the probability that the average time interval between arrivals of successive cars is more than 3 sec. Give the answer to four decimal places, if necessary.

Question 55

Find the value of the constant k so that the function is a probability density function in the indicated interval. ​ ​

A)
B) ​
C) ​
D) ​
E) ​

Question 56

Find the value of the constant k so that the function is a probability density function in the indicated interval.
​

Question 57

f is a joint probability density function on D. Find . ​ ​

A) ​
B) ​
C) ​
D) ​
E) ​

Question 58

National Welding Company uses industrial robots in some of its assembly-line operations. Management has determined that the lengths of time in hours between breakdowns are exponentially distributed with probability density function . ​
What is the probability that a robot selected at random will break down between 400 and 500 hr of use? Round your answer to the nearest hundredth.
​

A) 0.06
B) ​0.30
C) ​0.15
D) ​0.79
E) ​0.51

Question 59

Find the value of the constant ​k so that the function is a joint probability density function on D. ​ ​

A)
B) ​
C) ​
D) ​
E) ​

Question 60

The function f is a probability density function defined on the given interval. Find . ​ ​

A) ​
B) ​
C) ​
D) ​
E) ​

Question 61

Find the value of the constant k so that the function is a joint probability density function on D.
​

Question 62

Find the value of the constant k so that the function is a probability density function in the indicated interval.
​

Question 63

The amount of snowfall (in feet) in a remote region of Alaska in the month of January is a continuous random variable with probability density function .
​
Round your answers to four decimal places, if necessary.
​
Find the probability that the amount of snowfall will be between 1 and 3 ft.
​
__________
​
Find the probability that the amount of snowfall will be more than 1 ft.
​
__________

Question 64

National Welding Company uses industrial robots in some of its assembly-line operations. Management has determined that the lengths of time in hours between breakdowns are exponentially distributed with probability density function
​ ​
What is the probability that a robot selected at random will break down between 500 and 800 hr of use? Round the answer to the nearest hundredth.
​
__________
​
What is the probability that a robot will break down after 1,100 hr of use? Round the answer to the nearest hundredth.
​
__________

Question 65

Find the value of the constant k so that the function is a joint probability density function on D.
​

Question 66

Find the value of the constant k so that the function is a probability density function in the indicated interval.
​

Question 67

Find the value of the constant k so that the function is a probability density function in the indicated interval.
​

Question 68

Find the value of the constant k so that the function is a joint probability density function on D.
​