Deck 8: Continuous Probability Distributions

Question 1

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

A)They are symmetrical.
B)They are bell-shaped.
C)The area under the curve is 1.0.
D)They have the same height.

Answer 1

The area under the curve is 1.0.

Question 2

Given that the random variable X is normally distributed with a mean of 20 and a standard deviation of 7, P(28  X  30) is:

A)2
B)0.0507.
C)0.8729
D)0.9236

Answer 2

0.0507.

Question 3

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

A)0.3333.
B)0.5000.
C)0.6667.
D)0.0025.

Answer 3

0.0025.

Question 4

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?

A)0.92.
B)-0.05.
C)0.05.
D)-0.92.

Question 5

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

A)Uniform.
B)Normal.
C)Exponential.
D)Poisson.

Question 6

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 usually larger than the standard deviation.
D)a mean always larger than the standard deviation.

Question 7

If the random variable X is exponentially distributed with parameter  = 1.75, then P(1.5  X  3.8), up to 4 decimal places, is:

A)0.0711.
B)0.0473.
C)0.1184.
D)0.4739.

Question 8

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

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

Question 9

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

A)0.6321.
B)0.3679.
C)0.2500.
D)None of these choices are correct.

Question 10

What are the values of z that correspond to the P(-z  Z  z) equal to 0.4778?

A)±0.2389.
B)± 0.71.
C)±0.64.
D)None of these choices are correct.

Question 11

Given that X is a normal variable, which of the following statements is (are) true?

A)The variable X + 5 is also normally distributed.
B)The variable X - 5 is also normally distributed.
C)The variable 5X is also normally distributed.
D)All of these choices are correct.

Question 12

Given that Z is a standard normal random variable, P(-1.23  Z 1.89) is:

A)0.1903.
B)0.9706.
C)0.8907.
D)0.8613.

Question 13

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

A)is bell-shaped.
B)is symmetrical.
C)approaches infinity as x approaches zero.
D)approaches zero as x approaches infinity.

Question 14

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

A)0.125.
B)0.975.
C)0.625.
D)0.500.

Question 15

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

A)0.0037.
B)0.5037.
C)0.9963.
D)0.4963.

Question 16

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

A)P(Z $\geq$ z).
B)P(Z $\leq$ z).
C)P(0 $\leq$ Z $\leq$ z).
D)P(Z $\geq$ -z).

Question 17

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?

A)0.51.
B)-0.51.
C)0.86.
D)-0.86.

Question 18

Given that Z is a standard normal variable, the value z for which P(Z  z) = 0.2580 is:

A)0.70.
B)0.758.
C)-0.65.
D)0.242.

Question 19

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

A)1.0.
B)0.50.
C)0.25.
D)any value smaller than 1.0.

Question 20

If Z is a standard normal random variable, then P(-2.28  Z  -1.96 ) is:

A)0.0250.
B)0.0146.
C)−0.0146.
D)0.0137.

Question 21

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?

A)11.63.
B)61.63.
C)63.16.
D)16.31.

Question 22

If the random variable X is uniformly distributed between 40 and 60, then P(35  X  45) is:

A)1.0.
B)0.5.
C)0.25.
D)0.0.

Question 23

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 area under the curve at x.
D)the height of the function at x.

Question 24

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

A)The mean of X will be greater than the standard deviation of X.
B)The mean of X will be smaller than the standard deviation of X.
C)The mean of X will equal the standard deviation of X.
D)None of these choices are correct.

Question 25

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

A)It is a symmetrical distribution.
B)The mean is always zero.
C)The mean, median and mode are all equal.
D)All of these choices are correct.

Question 26

Which of the following is true for a normal distribution?

A)It is unimodal.
B)It is symmetrical.
C)It has a bell shape.
D)All of these choices are correct.

Question 27

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

A)binomial function.
B)normal function.
C)Poisson function.
D)probability density function.

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?

A)−2.92
B)1.96
C)4.92
D)None of these choices are correct.

Question 29

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

A)2.14
B)2.88
C)3
D)None of these choices are correct.

Question 30

Which of the following is not true for a random variable X that is uniformly distributed over the interval $a \leq x \leq b$ ?

A)E(X) = (a + b)/2.
B)V(X) = $( b - a ) ^ { 2 }$ /12.
C) $\sigma$ = (b - a)/6.
D) $f ( x ) = 1 / ( b - a )$ if a ? x ? b.

Question 31

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

A)any value between 1 and 11.
B)0.100.
C)0.091.
D)above 11.

Question 32

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

A)Binomial distribution.
B)Normal distribution.
C)Poisson distribution.
D)Uniform distribution.

Question 33

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

A)the value z is to the left of the mean.
B)the value z is to the right of the median.
C)the z value is to the right of the mean.
D)both B and C are correct.

Question 34

The expected value, E(X), of a uniform random variable X defined over the interval $a \leq x \leq b$ , is:

A)a + b.
B)a - b.
C)(a + b)/2.
D)(a - b)/2.

Question 35

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?

A)0.0300.
B)0.0668.
C)0.0968.
D)0.8364.

Question 36

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

A)1/(b - a).
B)1/(a - b).
C)(b - a)/2.
D)(a - b)/2.

Question 37

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

A)is different for various values of the random variable X.
B)is the same for various values of the random variable X.
C)increases as the values of the random variable X increase.
D)None of these choices are correct.

Question 38

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

A)mean and median of X are specified.
B)median and mode of X are specified.
C)mean and mode of X are specified.
D)mean and variance of X are specified.

Question 39

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 40

Which of the following distributions is not symmetrical?

A)Exponential
B)Uniform.
C)Normal.
D)None of these choices are correct.

Question 41

Which of the following is not true for an exponential distribution with parameter $\lambda$ ?

A) $\mu = 1 / \lambda$
B) $\sigma = 1 / \lambda$
C)The distribution is completely determined once the value of $\lambda$ is known.
D)The distribution is a two-parameter distribution, since the mean and standard deviation are equal.

Question 42

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

Question 43

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 44

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:

A) $n p \geq 5 \text { and } n ( 1 - p ) \geq 5$
B) $n p \leq 5 \text { and } n ( 1 - p ) \leq 5 \text {. }$
C) $n p \geq 5 \text { and } n ( 1 - p ) \leq 5$
D) $n p \leq 5 \text { and } n ( 1 - p ) \geq 5$

Question 45

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

Question 46

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

Question 47

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 48

In the normal distribution, the curve is skewed.

Question 49

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 50

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

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

Question 51

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

Question 52

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

A)x - 0.5.
B)x + 0.5.
C)x - 1.
D)x + 1.

Question 53

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

Question 54

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

Question 55

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

Question 56

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 57

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 58

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

A)smaller than the median.
B)larger than the mode.
C)always equal to zero.
D)always smaller than zero.

Question 59

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 60

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

A)0.1736.
B)0.45.
C)-0.1736.
D)-0.45.

Question 61

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

Question 62

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

Question 63

The normal approximation to the binomial distribution works best when the number of trials is large, and when the binomial distribution is symmetrical (like the normal).

Question 64

In the standard normal distribution, z0.05 = 1.645 means that there is a 5% chance that the standard normal random variable Z assumes a value above 1.645.

Question 65

If the random variable X is exponentially distributed and the parameter of the distribution  = 4, then P(X  0.25) = 0.3679.

Question 66

The exponential distribution is suitable to model the length of time that elapses before the first telephone call is received by a switchboard.

Question 67

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

Question 68

A certain brand of flood lamps has a lifetime that is normally distributed with a mean of 3750 hours and a standard deviation of 300 hours.
a. What proportion of these lamps will last for more than 4000 hours?
b. What lifetime should the manufacturer advertise for these lamps in order that only 2% of the lamps will burn out before the advertised lifetime?

Question 69

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

Question 70

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

Question 71

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.

Question 72

If X is a normal random variable with a mean of 78 and a standard deviation of 5, find the following probabilities:
a.
b.
c.

Question 73

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

Question 74

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

Question 75

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 76

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

Question 77

The mean of any normal distribution is always zero.

Question 78

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 79

The values of zA are the 100(1 - A)th percentiles of a standard normal random variable.

Question 80

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.