Deck 8: Continuous Probability Distributions

Question 1

A standard normal distribution is a normal distribution with:

Answer 1

A

Question 2

Given that Z is a standard normal variable, the value z for which P(Z $\le$ z) = 0.2580 is: $\begin{array}{|l|l|}\hline\text { A. } & 0.70 . \\\hline \text { B. } & 0.758 . \\\hline \text { C. } & -0.65 . \\\hline \text { D. } & 0.242 . \\\hline\end{array}$

Answer 2

C

Question 3

What are the values of z that correspond to the P(-z $\le$ Z $\le$ z) equal to 0.4778? $\begin{array}{|l|l|}\hline A&\pm 0.2389\\\hline B&\pm 0.71\\\hline C&\pm 0.64 \\\hline D&\text { None of these choices are correct.}\\\hline \end{array}$

Answer 3

C

Question 4

If Z is a standard normal random variable, then P(-2.28 $\le$ Z $\le$ -1.96 ) is: $\begin{array}{|l|l|}\hline \text { A. } & 0.0250 . \\\hline \text { B. } & 0.0146 . \\\hline \text { C. } & -0.0104 . \\\hline \text { D. } & 0.0 .0104 . \\\hline\end{array}$

Question 5

If the random variable X is normally distributed with a mean of 75 and a standard deviation of 8, then P(X ≤ 75) is:

Question 6

Given that X is a normal variable, which of the following statements is (are) true? $\begin{array}{|l|l|}\hline A&\text {The variable \( X+5 \) is also normally distributed. }\\\hline B&\text {The variable \( X-5 \) is also normally distributed. }\\\hline C&\text { The variable \( 5 X \) is also normally distributed. }\\\hline D&\text {All of these choices are correct. }\\\hline \end{array}$

Question 7

If the random variable X is exponentially distributed with parameter = 4, then the probability P(X 0.25), up to 4 decimal places, is:

Question 8

What proportion of the data from a normal distribution is within 2 standard deviations of the mean?

Question 9

Given that Z is a standard normal random variable, P(Z > − 2.68) is:

Question 10

Like the normal distribution, the exponential density function f(x):

Question 11

Given that Z is a standard normal random variable, what is the value of Z if the area to the right of Z is 0.8212?

Question 12

If the continuous random variable X is uniformly distributed over the interval [15, 25], then the mean of X is:

Question 13

Which of the following is always true for all probability density functions of continuous random variables? $\begin{array}{|l|l|}\hline A&\text {They are symmetrical. }\\\hline B&\text { They are bell-shaped.}\\\hline C&\text {The area under the curve is 1.0 . }\\\hline D&\text {They have the same height. }\\\hline \end{array}$

Question 14

Given that Z is a standard normal random variable, what is the value of Z if the area to the left of Z is 0.1949?

Question 15

If the random variable X is exponentially distributed with parameter $\lambda$ = 1.75, then P(1.5 $\le$ X $\le$ 3.8), up to 4 decimal places, is: $\begin{array}{|l|l|}\hline\text { A. } & 0.0711 . \\\hline \text { B. } & 0.0473 . \\\hline \text { C. } & 0.1184 . \\\hline \text { D. } & 0.4739 \\\hline\end{array}$

Question 16

Given that Z is a standard normal random variable, P(-1.23 $\le$ Z $\le$ 1.89) is: $\begin{array}{|l|l|}\hline\text { A. } & 0.1903 . \\\hline \text { B. } & 0.9706 . \\\hline \text { C. } & 0.8907 . \\\hline \text { D. } & 0.7803 . \\\hline\end{array}$

Question 17

If the random variable X is exponentially distributed with parameter = 3, then P(X 2), up to 4 decimal places, is:

Question 18

Given that the random variable X is normally distributed with a mean of 20 and a standard deviation of 7, P(28 $\le$ X $\le$ 30) is: $\begin{array}{|l|l|}\hline \text { A. } & 2 \\\hline \text { B. } & 0.0507 . \\\hline \text { C. } & 0.8729 \\\hline \text { D. } & 0.9236 \\\hline\end{array}$

Question 19

If X is a normal random variable with a standard deviation of 10, then 3X has a standard deviation equal to:

Question 20

Which of the following distributions is suitable to model the length of time that elapses before the first telephone call is received by a switchboard?

Question 21

The mean of the exponential distribution equals the mean of the Poisson distribution only when the former distribution has a mean equal to:

Question 22

Which of the following is true for a normal distribution? $\begin{array}{|l|l|}\hline A&\text {It is unimodal. }\\\hline B&\text {It is symmetrical. }\\\hline C&\text {It has a bell shape. }\\\hline D&\text {All of these choices are correct. }\\\hline \end{array}$

Question 23

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

Question 24

The expected value, E(X), of a uniform random variable X defined over the interval , is:

Question 25

The probability density function f(x) for a uniform random variable X defined over the interval [1, 11] is:

Question 26

If the random variable X is exponentially distributed, then which of the following statements best describes the mean of X?

Question 27

Find the value of σ if it is know that X is normally distributed with mean 5 and 14.92% of the values are above 8?

Question 28

Find the value of µ, if X is a normal random variable, with standard deviation 2, and 2.5% of the values are below 1?

Question 29

If the random variable X is uniformly distributed between 40 and 60, then P(35 $\le$ X $\le$ 45) is: $\begin{array}{|l|l|}\hline\text { A. } & 1.0 . \\\hline \text { B. } & 0.5 . \\\hline \text { C. } & 0.25 \\\hline \text { D. } & 0.0 . \\\hline\end{array}$

Question 30

The probability density function f(x) of a random variable X that is uniformly distributed between a and b is:

Question 31

If Z is a standard normal random variable, the area between z = 0.0 and z =1.30 is 0.4032, while the area between z = 0.0 and z = 1.50 is 0.4332. What is the area between z = -1.30 and z = 1.50?

Question 32

Which of the following distributions is not symmetrical?

Question 33

The probability density function f(x) of a random variable X that is normally distributed is completely determined once the:

Question 34

Which of the following is not true for a random variable X that is uniformly distributed over the interval $a \leq x \leq b$ ? $\begin{array}{|l|l|}\hline A&E(X)=(a+b) / 2 \\\hline B&V(X)=(b-a)^{2} / 12\\\hline C&\sigma=(b-a) / 6\\\hline D&f(x)=\frac{1}{b-a} \text { if } a \leq x \leq b\\\hline \end{array}$

Question 35

If the z-value for a given value x of the random variable X is z = 2.326, and the distribution of X is normal with a mean of 50 and a standard deviation of 5, to what x-value does this z-value correspond?

Question 36

Which of the following distributions is considered the cornerstone distribution of statistical inference?

Question 37

The height of the function for a uniform probability density function f(x):

Question 38

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

Question 39

The function f(x) that defines the probability distribution of a continuous random variable X is a:

Question 40

Which of the following distributions is appropriate to measure the length of time between arrivals at a grocery checkout counter?

Question 41

Given that Z is a standard normal variable, the value z for which P(Z z) = 0.6736 is:

Question 42

If the mean of an exponential distribution is 4, then the value of the parameter is:

Question 43

Which of the following is not true for an exponential distribution with parameter $\lambda$ ? $\begin{array}{|l|l|}\hline A& \mu=1 / \lambda\\\hline B&\sigma=1 / \lambda\\\hline C&\text { The distribution is completely determined once the value of \( \lambda \) is known.}\\\hline D&\text {The distribution is a two-parameter distribution, since the }\\&\text { mean and standard deviation are equal.}\\\hline \end{array}$

Question 44

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

Question 45

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

Question 46

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

Question 47

In the normal distribution, the flatter the curve, the larger the standard deviation

Question 48

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

Question 49

Given that X is a binomial random variable, the binomial probability P(X x) is approximated by the area under a normal curve to the right of:

Question 50

In the exponential distribution, the value of x can be any of an infinite number of values in the given range.

Question 51

In the normal distribution, the curve is skewed.

Question 52

Which of the following is a characteristic of a normal distribution?

Question 53

In the normal distribution, the total area under the curve is equal to one.

Question 54

In the normal distribution, the curve is asymptotic but never intercepts the horizontal axis either to the left or right.

Question 55

Given that X is a binomial random variable, the binomial probability P(X $\le$ x) is approximated by the area under a normal curve to the left of: $\begin{array}{|l|l|}\hline\text { A. } & x . \\\hline \text { B. } & -x . \\\hline \text { C. } & x+0.5 . \\\hline \text { D. } & x-0.5 . \\\hline\end{array}$

Question 56

Suppose that the probability p of a success on any trial of a binomial distribution equals 0.80. For which value of the number of trials, n, would the normal distribution provide a good approximation to the binomial distribution?

Question 57

Given a binomial distribution with n trials and probability p of a success on any trial, a conventional rule of thumb is that the normal distribution will provide an adequate approximation of the binomial distribution if:

Question 58

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

Question 59

In the normal distribution, the mean, median and mode are all at the same position on the horizontal axis since the distribution is symmetric.

Question 60

Given that Z is a standard normal random variable, a positive z value means that:

Question 61

If the random variable X is exponentially distributed and the parameter of the distribution $\lambda$ = 4, then P(X $\le$ 0.25) = 0.3679.

Question 62

For a normal curve, if the mean is 25 minutes and the standard deviation is 5 minutes, the area to the right of 25 minutes is 0.50.

Question 63

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

Question 64

A random variable X is normally distributed with a mean of 150 and a variance of 25. Given that
X = 120, its corresponding z-score is 6.0.

Question 65

Using the standard normal curve, the z-score representing the 90th percentile is 1.28.

Question 66

A random variable X is standardised when each value of X has the mean of X subtracted from it, and the difference is divided by the standard deviation of X.

Question 67

The mean and standard deviation of an exponential random variable cannot equal to each other.

Question 68

The mean and standard deviation of a normally distributed random variable that has been standardised are one and zero, respectively.

Question 69

The mean of any normal distribution is always zero.

Question 70

Using the standard normal curve, the z-score representing the 10th percentile is 1.28.

Question 71

The mean and the standard deviation of an exponential distribution are equal to each other.

Question 72

Using the standard normal curve, the z-score representing the 75th percentile is 0.75.

Question 73

Using the standard normal curve, the probability or area between z = -1.28 and z = 1.28 is 0.1003

Question 74

If the random variable X is exponentially distributed with $\lambda$ = 2 parameter, then the variance of the distribution is 0.5.

Question 75

For a normal curve, if the mean is 20 minutes and the standard deviation is 5 minutes, the area to the right of 13 minutes is 0.9192.

Question 76

Using the standard normal curve, the area between z = 0 and z = 3.50 is about 0.50.

Question 77

Let z₁ be a z-score that is unknown but identifiable by position and area. If the symmetrical area between -z₁ and + z₁ is 0.9544, the value of z₁ is 2.0.

Question 78

Let z₁ be a z-score that is unknown but identifiable by position and area. If the area to the right of z₁ is 0.7291, the value of z₁ is -0.61.

Question 79

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

Question 80

A random variable X is normally distributed with a mean of 250 and a standard deviation of 50. Given that X = 175, its corresponding z-score is -1.50.