Deck 7: Continuous Probability Distributions

Question 1

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.

Answer 1

False

Question 2

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

Answer 2

True

Question 3

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

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

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 6

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

Question 7

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

Question 8

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

Question 9

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

Question 10

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

Question 11

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

Question 12

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 13

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

Question 14

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

Question 15

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 very large.

Question 16

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

Question 17

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

Question 18

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 19

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

Question 20

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 21

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

Question 22

The variance of a Student t random variable with v degrees of freedom (v > 2) is always greater than 1.

Question 23

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

Question 24

Like that of the Student t distribution, the shape of the chi-squared distribution depends on its number of degrees of freedom.

Question 25

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

Question 26

The value of $\chi$ ² with v degrees of freedom such that the area to its right under the chi-squared curve is equal to A is denoted by
, while
denotes the value such that the area to its left is A.

Question 27

We define as the value of the F with v₁ and v₂ degrees of freedom such that the area to its right under the F curve is A, while is defined as the value such that the area to its left is A.

Question 28

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

Question 29

The value of A such that P(-A $\le$ t $\le$ A) = 0.95, where the degrees of freedom are 20, is 2.086.

Question 30

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

Question 31

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

Question 32

The expected value of the Student t distribution is zero.

Question 33

The variance of a $\chi$ ² distribution is twice the value of its mean.

Question 34

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 35

As the degrees of freedom approach infinity, the values of a Student t distribution approach those of a standard normal distribution.

Question 36

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 37

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 38

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 39

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

Question 40

The value of an F distribution with v₁ = 5 and v₂ = 10 degrees of freedom such that the area to its left is 0.95 is 3.33.

Question 41

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 42

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 43

The Student t distribution looks similar in shape to a standard normal distribution, except it is not as widely spread.

Question 44

The value of a chi-squared distribution with 8 degrees of freedom such that the area to its left is 0.95 is 2.73.

Question 45

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 46

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 47

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 48

Suppose f(x) = 1/4 over the range a $\le$ x $\le$ 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 49

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 50

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 51

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 52

The variance of a Student t distribution approaches zero as the degrees of freedom approaches infinity.

Question 53

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 54

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 55

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 56

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 57

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 58

The value of an F distribution with v₁ = 6 and v₂ = 9 degrees of freedom such that the area to its right is 0.05 is 3.37.

Question 59

The value of a chi-squared distribution with 5 degrees of freedom such that the area to its left is 0.10 is 1.61.

Question 60

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 61

What number corresponds to F_0.95,4,8?

A) 6.040
B) 3.840
C) 0.260
D) 0.166

Question 62

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 63

Which of the following statements is false?

A) The chi-squared distribution is positively skewed.
B) All the values of the chi-squared distribution are non-negative.
C) The shape of the chi-squared distribution depends on its degrees of freedom.
D) All of these choices are true.

Question 64

If , then the number of degrees of freedom v is:

A) 5
B) 6
C) 7
D) 8

Question 65

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 66

Suppose X has a chi-squared distribution with 10 degrees of freedom. The variance of X is:

A) 20
B) 10
C) 9
D) None of these choices.

Question 67

The Student t distribution with parameter v = 4 has a variance V(t) equal to:

A) 4
B) 0
C) 2
D) 1

Question 68

What number corresponds to t_0.05,10?

A) 1.812
B) 1.372
C) 2.228
D) 1.833

Question 69

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 70

What number corresponds to F_.025,3,5?

A) 14.88
B) 7.76
C) 12.06
D) 5.41

Question 71

Suppose X has a chi-squared distribution with 10 degrees of freedom. The mean of X is:

A) 10
B) 9
C) 20
D) None of these choices.

Question 72

The Student t distribution with parameter v = 2 has a mean E(t) equal to:

A) 0
B) 1
C) 2
D) None of these choices.

Question 73

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 74

Which of the following statements is correct regarding the percentile points of the F distribution?

A) F_0.10,10,20 = 1/F_0.90,10,20
B) F_0.90,10,20 = 1/F_0.10,20,10
C) F_0.90,10,20 = 1/F_0.90,20,10
D) F_0.10,10,20 = 1/F_0.10,20,10

Question 75

The Student t distribution:

A) is symmetrical.
B) approaches the normal distribution as the degrees of freedom increase.
C) has more area in the tails than the standard normal distribution does.
D) All of these choices are true.

Question 76

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 77

What number corresponds to ?

A) 28.30
B) 26.22
C) 21.00
D) 5.23

Question 78

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 79

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 80

If P(t > t_.01,v) = 2.50, then the number of degrees of freedom v is:

A) 20
B) 21
C) 22
D) 23