Deck 8: Continuous Probability Distributions

Question 1

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.

Answer 1

All of these choices are true.

Question 2

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

Answer 2

False

Question 3

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.

Answer 3

True

Question 4

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 5

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 6

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 7

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 8

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

Question 9

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

Question 10

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

Question 11

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 12

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.

Question 13

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

Question 14

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

Question 15

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

Question 16

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

Question 17

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

Question 18

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

Question 19

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

Question 20

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

Question 21

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 22

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

Question 23

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

Question 24

Electronics Test
The time it takes a student to finish a electronics test has a uniform distribution between 50 and 70 minutes.

-{Electronics Test Narrative} What is the probability density function for this uniform distribution?

Question 25

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 26

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 27

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

Question 28

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 29

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 30

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 31

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

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

Question 32

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 33

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

Question 34

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

Question 35

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 36

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

Question 37

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

Question 38

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

Question 39

Electronics Test
The time it takes a student to finish a electronics test has a uniform distribution 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 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

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 42

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

Question 43

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 44

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

Question 45

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 46

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 47

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

Question 48

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 49

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

Question 50

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 51

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

Question 52

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

Question 53

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

Question 54

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

Question 55

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 56

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 57

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 58

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

Question 59

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 60

Electronics Test
The time it takes a student to finish a electronics test has a uniform distribution 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 61

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 62

Suppose X has a normal distribution with mean 10 and standard deviation 2.The probability that X is less than 8 is equal to the probability that Z is less than ____________________.

Question 63

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 64

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

Question 65

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.

Question 66

Suppose X has a normal distribution with mean 40 and standard deviation 2.Shifting all the X values to the right 10 units results in a normal distribution with mean ____________________ and standard deviation ____________________.

Question 67

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 68

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 69

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 70

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 71

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

A) P(Z $\ge$ z)
B) P(Z $\le$ z)
C) P(0 $\le$ Z $\le$ z)
D) P(Z $\ge$ -z)

Question 72

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 73

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 74

____________________ the value of $\sigma$ in a normal distribution will make it wider.

Question 75

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 76

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 77

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 78

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 79

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

Question 80

We standardize a random variable by subtracting its ____________________ and dividing by its ____________________.