Question 1

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.

Accepted Answer

We can use the formula for standardizing a normal random variable:

z = (x - μ) / σ

Rearranging the formula to solve for x, we get:

x = zσ + μ

Substituting the given values, we get:

x = (2.326)(5) + 50 = 61.63

Therefore, the z-value of 2.326 corresponds to an x-value of 61.63.

Question 2

If the random variable X is uniformly distributed between 40 and 60, then P(35 $\le$  X $\le$  45) is:
A)1.0.
B)0.5.
C)0.25.
D)0.0.

Accepted Answer

The probability is calculated based on the range of values that X can take. Since X is uniformly distributed between 40 and 60, the total range is 20 units. The probability of X being between 35 and 45 is the same as it being between 40 and 45 (since values below 40 are not possible for X), which is a range of 5 units. Therefore, the probability is 5/20 = 0.25.

Question 3

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 $\le$  x $\le$  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.

Accepted Answer

The probability density function (pdf) for a continuous random variable X represents the relative likelihood of X taking values in a given interval, rather than the probability that X takes on a specific value. Therefore, choice A is incorrect. The area under the curve of the pdf represents the probability of X taking values within a specific interval, not at a specific value of X, so choice C is also incorrect. The pdf represents the height of the function at a given value of X, so choice D is correct.

Question 4

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.

Accepted Answer

For an exponentially distributed random variable X, the mean and the standard deviation are equal. This is a characteristic property of the exponential distribution.

Question 5

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.

Accepted Answer

A standard normal distribution is a distribution with a mean of 0 and a standard deviation of 1. It is also a symmetrical distribution, which means that the mean, median, and mode are all equal. Therefore, all of the choices are correct.

Question 6

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.

Accepted Answer

All of the given choices are true for a normal distribution. It is unimodal, symmetrical, and has a bell shape.

Question 7

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.

Accepted Answer

The probability distribution of a continuous random variable is defined by a probability density function (PDF), which is a function that assigns probabilities to intervals of values of X. The PDF of a continuous random variable cannot be a binomial, Poisson, or normal function, as those are specific distributions for discrete random variables or normal continuous distributions. Therefore, the best choice is D, probability density function.

Question 8

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.

Accepted Answer

Using the standard normal distribution table, we can find that the z-score corresponding to the 2.5th percentile is -1.96. 
Therefore, we have:

1 = µ + (-1.96)(2)
 
Solving for µ, we get:

µ = 1 + 3.92 = 4.92

Question 9

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.

Accepted Answer

The answer of Find the value of σ if it...

Question 10

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 $\le$  x $\le$  b.

Accepted Answer

C) is not true because the correct formula for the standard deviation ($\sigma$) of a uniform distribution is $\sigma = \sqrt{(b - a)^2 / 12}$, not $(b - a)/6$.

Question 11

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.

Accepted Answer

The probability density function (PDF) for a uniform distribution on the interval [a, b] is given by 1/(b-a). Here, a = 1 and b = 11, so the PDF is 1/(11-1) = 1/10 = 0.100.

Question 12

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.

Accepted Answer

The normal distribution is considered the cornerstone distribution of statistical inference because of the central limit theorem. This theorem states that the sampling distribution of the mean of any independent, random variable will approximate a normal distribution as the sample size increases, regardless of the shape of the population distribution. This allows for the use of many inferential statistical tests and makes the normal distribution important in many fields of study.

Question 13

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.

Accepted Answer

A positive z value in a standard normal distribution indicates that the value is to the right of the mean, which is also the median in a symmetric distribution like the normal distribution.

Question 14

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.

Accepted Answer

The expected value of a uniform random variable defined over the interval $a \leq x \leq b$ is given by $\frac{a + b}{2}$.

Question 15

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.

Accepted Answer

The area between z = -1.30 and z = 1.50 is found by adding the area from z = 0 to z = 1.30 (0.4032) and the area from z = 0 to z = 1.50 (0.4332), then subtracting the overlap (the area from z = 0 to z = 1.30, since it's included in both calculations). However, since the normal distribution is symmetric, the area from z = 0 to z = -1.30 is also 0.4032. Therefore, the total area is 0.4032 (from z = -1.30 to z = 0) + 0.4332 (from z = 0 to z = 1.50) = 0.8364.

Question 16

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.

Accepted Answer

The probability density function of a continuous uniform distribution is a constant k between the limits a and b, and zero elsewhere. To find k, we use the fact that the total area under the PDF must be equal to 1 (since the probability of X being between a and b is 1). Therefore, k times the width of the interval (b-a) must be equal to 1. Solving for k, we get k = 1/(b-a), which is choice A.

Question 17

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.

Accepted Answer

A uniform probability density function has a constant height for all values of the random variable X, hence the height is the same for various values of X.

Question 18

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.

Accepted Answer

The probability density function f(x) of a normal distribution is completely determined by its mean and variance. The mean determines the location of the peak of the curve, while the variance determines the spread of the curve. The median and mode may be different from the mean for a normal distribution, but they do not determine the shape of the curve. Therefore, the correct answer is D, mean and variance of X are specified.

Question 19

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.

Accepted Answer

A standard normal variable has a mean of 0 and a standard deviation of 1. The variance is the square of the standard deviation, so the variance of Z is always equal to (1^2) = 1.

Question 20

Which of the following distributions is not symmetrical?
A)Exponential
B)Uniform.
C)Normal.
D)None of these choices are correct.

Accepted Answer

The exponential distribution is not symmetrical because it has a long tail on one side, indicating that the probability of getting extreme values is higher on one side than the other. The uniform and normal distributions are both symmetrical, with the uniform having a constant probability of getting any value within its range, and the normal having a bell-shaped curve with equal probabilities on either side of the mean. Therefore, D is not the correct answer.

Quiz 8: Continuous Probability Distributions

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?

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

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

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

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

Which of the following is true for a normal distribution?

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

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?

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?

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

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

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

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

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

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?

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

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

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

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

Which of the following distributions is not symmetrical?