Deck 8: Continuous Probability Distributions

Question 1

Which of the following represents a difference between continuous and discrete random variables?

A)Continuous random variables assume an uncountable number of values,and discrete random variables do not.
B)The probability for any individual value of a continuous random variable is zero,but for discrete random variables it is not.
C)Probability for continuous random variables means finding the area under a curve,while for discrete random variables it means summing individual probabilities.
D)All of these choices are true.

Answer 1

All of these choices are true.

Question 2

A probability density function shows the probability for each value of X.

Answer 2

False

Question 3

To be a legitimate probability density function,all possible values of f(x)must lie between 0 and 1 (inclusive).

Answer 3

True

Question 4

In practice,we frequently use a continuous distribution to approximate a discrete one when the number of values the variable can assume is countable but large.

Question 5

A continuous random variable X has a uniform distribution between 5 and 25 (inclusive),then P(X = 15)= 0.05.

Question 6

If X is a continuous random variable on the interval [0,10],then P(X > 5)= P(X ≥ 5).

Question 7

Continuous probability distributions describe probabilities associated with random variables that are able to assume any finite number of values along an interval.

Question 8

If X is a continuous random variable on the interval [0,10],then P(X = 5)= f(5)= 1/10.

Question 9

Since there is an infinite number of values a continuous random variable can assume,the probability of each individual value is virtually 0.

Question 10

To be a legitimate probability density function,all possible values of f(x)must be non-negative.

Question 11

If a point y lies outside the range of the possible values of a random variable X,then f(y)must equal zero.

Question 12

A continuous random variable X has a uniform distribution between 10 and 20 (inclusive),then the probability that X falls between 12 and 15 is 0.30.

Question 13

A continuous random variable is one that can assume an uncountable number of values.

Question 14

A continuous probability distribution represents a random variable having an infinite number of outcomes which may assume any number of values within an interval.

Question 15

The probability density function,f(x),for any continuous random variable X,represents:

A)all possible values that X will assume within some interval a ≤ x ≤ b
B) the probability that X takes on a specific value x.
C)The height of the density function at x.
D) None of these choices.

Question 16

The sum of all values of f(x)over the range of [a,b] must equal one.

Question 17

Let X represent weekly income expressed in dollars.Since there is no set upper limit,we cannot identify (and thus cannot count)all the possible values.Consequently,weekly income is regarded as a continuous random variable.

Question 18

Which of the following is always true for all probability density functions of continuous random variables?

A)The probability at any single point is zero.
B)They contain an uncountable number of possible values.
C)The total area under the density function f(x)equals 1.
D)All of these choices are true.

Question 19

We distinguish between discrete and continuous random variables by noting whether the number of possible values is countable or uncountable.

Question 20

A continuous random variable X has a uniform distribution between 5 and 15 (inclusive),then the probability that X falls between 10 and 20 is 1.0.

Question 21

For a continuous random variable,the probability for each individual value of X is ____________________.

Question 22

A(n)____________________ random variable is one that assumes an uncountable number of possible values.

Question 23

You can use a continuous random variable to ____________________ a discrete random variable that takes on a countable,but very large,number of possible values.

Question 24

A(n)____________________ random variable has a density function that looks like a rectangle and you can use areas of a rectangle to find probabilities for it.

Question 25

The probability density function of a uniform random variable on the interval [0,5] must be ____________________ for 0 ≤ x ≤ 5.

Question 26

The probability density function f(x)for a uniform random variable X defined over the interval [2,10] is

A)0.20
B)8
C)4
D)None of these choices.

Question 27

A continuous random variable X has the following probability density function:
f(x)= 1/4,0 ≤ x ≤ 4
Find the following probabilities:
a.P(X ≤ 1)
b.P(X ≥ 2)
c.P(1 ≤ X ≤ 2)
d.P(X = 3)

Question 28

Probability for continuous random variables is found by finding the ____________________ under a curve.

Question 29

To find the probability for a uniform random variable you take the ____________________ times the ____________________ of its corresponding rectangle.

Question 30

The total area under f(x)for a continuous random variable must equal ____________________.

Question 31

Which of the following is true about f(x)when X has a uniform distribution over the interval [a,b]?

A)The values of f(x)are different for various values of the random variable X.
B) f(x)equals one for each possible value of X.
C)F(x)equals one divided by the length of the interval from a to b.
D) None of these choices.

Question 32

Suppose f(x)= 1/4 over the range a ≤ x ≤ b,and suppose P(X > 4)= 1/2.What are the values for a and b?

A)0 and 4
B)2 and 6
C)Can be any range of x values whose length (b − a)equals 4.
D)Cannot answer with the information given.

Question 33

Which of the following does not represent a continuous uniform random variable?

A)f(x)= 1/2 for x between −1 and 1,inclusive.
B)f(x)= 10 for x between 0 and 1/10,inclusive.
C)f(x)= 1/3 for x = 4,5,6.
D)None of these choices represents a continuous uniform random variable.

Question 34

Waiting Time
The length of time patients must wait to see a doctor at an emergency room in a large hospital has a uniform distribution between 40 minutes and 3 hours. ​ ​
{Waiting Time Narrative} What is the probability density function for this uniform distribution?

Question 35

The probability density function f(x)of a random variable X that has a uniform distribution between a and b is

A)(b + a)/2
B)1/b − 1/a
C)(a − b)/2
D)None of these choices.

Question 36

Waiting Time
The length of time patients must wait to see a doctor at an emergency room in a large hospital has a uniform distribution between 40 minutes and 3 hours. ​ ​
{Waiting Time Narrative} What is the probability that a patient would have to wait exactly one hour?

Question 37

Suppose X is a continuous random variable for X between a and
b.
b.Then its probability ____________________ function must non-negative for all values of X between a and

Question 38

Suppose f(x)= 0.25.What range of possible values can X take on and still have the density function be legitimate?

A)[0,4]
B)[4,8]
C)[−2,+2]
D)All of these choices are true.

Question 39

If the random variable X has a uniform distribution between 40 and 50,then P(35 ≤ X ≤ 45)is:

A)1.0
B)0.5
C)0.1
D)undefined.

Question 40

Waiting Time
The length of time patients must wait to see a doctor at an emergency room in a large hospital has a uniform distribution between 40 minutes and 3 hours. ​ ​
{Waiting Time Narrative} What is the probability that a patient would have to wait between one and two hours?

Question 41

Electronics Test
the time it takes a student to finish a electronics test has a uniform distrubtion between 50 and 70 minutes.
{Electronics Test Narrative} Find the probability that a student will take more than 60 minutes to finish the test.

Question 42

Subway Waiting Time
At a subway station the waiting time for a subway is found to be uniformly distributed between 1 and 5 minutes. ​ ​
{Subway Waiting Time Narrative} What is the probability that the subway arrives in the first minute and a half?

Question 43

A random variable X has a normal distribution with a mean of 250 and a standard deviation of 50.Given that X = 175,its corresponding value of Z is −1.50.

Question 44

Subway Waiting Time
At a subway station the waiting time for a subway is found to be uniformly distributed between 1 and 5 minutes. ​ ​
{Subway Waiting Time Narrative} What is the probability of waiting no more than 3 minutes?

Question 45

The time required to complete a particular assembly operation has a uniform distribution between 25 and 50 minutes.
a.What is the probability density function for this uniform distribution?
b.What is the probability that the assembly operation will require more than 40 minutes to complete?
c.Suppose more time was allowed to complete the operation,and the values of X were extended to the range from 25 to 60 minutes.What would f(x)be in this case?

Question 46

A national standardized testing company can tell you your relative standing on an exam without divulging the mean or the standard deviation of the exam scores.

Question 47

Subway Waiting Time
At a subway station the waiting time for a subway is found to be uniformly distributed between 1 and 5 minutes. ​ ​
{Subway Waiting Time Narrative} What is the probability density function for this uniform distribution?

Question 48

A random variable X is standardized by subtracting the mean and dividing by the variance.

Question 49

Electronics Test
the time it takes a student to finish a electronics test has a uniform distrubtion between 50 and 70 minutes.
{Electronics Test Narrative} What is the median amount of time it takes a student to finish the test?

Question 50

Electronics Test
the time it takes a student to finish a electronics test has a uniform distrubtion between 50 and 70 minutes.
{Electronics Test Narrative} What is the mean amount of time it takes a student to finish the test?

Question 51

Electronics Test
the time it takes a student to finish a electronics test has a uniform distrubtion between 50 and 70 minutes.
{Electronics Test Narrative} Find the probability that a student will take no less than 55 minutes to finish the test.

Question 52

Given that Z is a standard normal random variable,a negative value of Z indicates that the standard deviation of Z is negative.

Question 53

A random variable X has a normal distribution with mean 132 and variance 36.If x = 120,its corresponding value of Z is 2.0.

Question 54

A normal distribution is symmetric; therefore the probability of being below the mean is 0.50 and the probability of being above the mean is 0.50.

Question 55

Electronics Test
the time it takes a student to finish a electronics test has a uniform distrubtion between 50 and 70 minutes.
{Electronics Test Narrative} What is the probability density function for this uniform distribution?

Question 56

Waiting Time
The length of time patients must wait to see a doctor at an emergency room in a large hospital has a uniform distribution between 40 minutes and 3 hours. ​ ​
{Waiting Time Narrative} What is the probability that a patient would have to wait no more than one hour?

Question 57

If we standardize the normal curve,we express the original X values in terms of their number of standard deviations away from the mean.

Question 58

If your golf score is 3 standard deviations below the mean,its corresponding value on the Z distribution is −3.

Question 59

Electronics Test
the time it takes a student to finish a electronics test has a uniform distrubtion between 50 and 70 minutes.
{Electronics Test Narrative} Find the probability that a student will take exactly one hour to finish the test.

Question 60

Subway Waiting Time
At a subway station the waiting time for a subway is found to be uniformly distributed between 1 and 5 minutes. ​ ​
{Subway Waiting Time Narrative} What is the median waiting time for this subway?

Question 61

Suppose Lamont's exam score was at the 80^th percentile on an exam whose mean was 90.What was Lamont's exam score?

A)76.81
B)72.00
C)80.00
D)Cannot tell without more information.

Question 62

Given that Z is a standard normal random variable,a negative value (z)on its distribution would indicate:

A)z is to the left of the mean.
B)the standard deviation of this Z distribution is negative.
C)the area between zero and the value z is negative.
D)None of these choices.

Question 63

Most values of a standard normal distribution lie between:

A)0 and 1
B)−3 and 3
C)0 and 3
D)minus infinity and plus infinity

Question 64

If X has a normal distribution with mean 60 and standard deviation 6,which value of X corresponds with the value z = 1.96?

A)x = 71.76
B)x = 67.96
C)x = 61.96
D)x = 48.24

Question 65

Tanner took a statistics test whose mean was 80 and standard deviation was 5.The total points possible was 100.Tanner's score was 2 standard deviations below the mean.What was Tanner's score,rounded to the nearest whole number?

A)78
B)70
C)90
D)None of these choices.

Question 66

The probability that a standard normal random variable Z is less than −3.5 is approximately 0.

Question 67

In the standard normal distribution,z_0.05 = 1.645 means that 5% of all values of z are below 1.645 and 95% are above it.

Question 68

If the value of Z is z = 99,that means you are at the 99th percentile on the Z distribution.

Question 69

Lamont took a psychology exam whose mean was 70 with standard deviation 5.He also took a calculus exam whose mean was 80 with standard deviation 10.He scored 85 on both exams.On which exam did he do better compared to the other students who took the exam?

A)He did better on the psychology exam,comparatively speaking.
B)He did better on the calculus exam,comparatively speaking.
C)He did the same on both exams,relatively speaking.
D)Cannot tell without more information.

Question 70

A larger standard deviation of a normal distribution indicates that the distribution becomes:

A)narrower and more peaked.
B)flatter and wider.
C)more skewed to the right.
D)more skewed to the left.

Question 71

Suppose X has a normal distribution with mean 70 and standard deviation 5.The 50th percentile of X is 70.

Question 72

The probability that Z is less than −2 is the same as one minus the probability that Z is greater than +2.

Question 73

The 10th percentile of a Z distribution has 10% of the Z-values lying above it.

Question 74

Stacy took a math test whose mean was 70 and standard deviation was 5.The total points possible was 100.Stacey's results were reported to be at the 95th percentile.What was Stacey's actual exam score,rounded to the nearest whole number?

A)95
B)78
C)75
D)62

Question 75

Given that Z is a standard normal random variable,the area to the left of a value z is expressed as

A)P(Z ≥ z)
B)P(Z ≤ z)
C)P(0 ≤ Z ≤ z)
D)P(Z ≥ −z)

Question 76

What proportion of the data from a normal distribution is within two standard deviations from the mean?

A)0.3413
B)0.4772
C)0.6826
D)0.9544

Question 77

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.

Question 78

Which of the following is not a characteristic for a normal distribution?

A)It is symmetrical.
B)The mean is always zero.
C)The mean,median,and mode are all equal.
D)It is a bell-shaped distribution.

Question 79

A standard normal distribution is a normal distribution with:

A)a mean of zero and a standard deviation of one.
B)a mean of one and a standard deviation of zero.
C)a mean always larger than the standard deviation.
D)None of these choices.

Question 80

Given that Z is a standard normal variable,the variance of Z:

A)is always greater than 2.0.
B)is always greater than 1.0.
C)is always equal to 1.0.
D)cannot assume a specific value.