Question 1

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?

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

Accepted Answer

First, we convert 8 to a z-score, using the formula

z = (x - μ) / σ

z = (8 - 5) / σ

z = 1.96

From standard normal tables, we know that the area above a z-score of 1.96 is 0.025. Since we want the area below 1.96 (since only 14.92% of values are above 8), we subtract 0.025 from 1, giving us 0.975.

Now we need to find the z-score that corresponds to the 97.5th percentile, which we can also look up in the standard normal tables. We find that it is 1.96.

So we have:

1.96 = (8 - 5) / σ

Solving for σ, we get:

σ = (8 - 5) / 1.96 = 1.53

However, this is not one of the answer choices. We need to round to the nearest hundredth, which gives us 1.52. But again, this is not one of the answer choices. The closest one is choice B, 2.88, but this does not seem to be derived from any calculation presented here. Therefore, the question may be flawed or incomplete, and there may not be a correct answer choice.

Question 2

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

For a uniform random variable, the probability density function is given by f(x)=$\frac{1}{b-a}$ where a and b are the lower and upper bounds of the interval. Therefore, for X defined over [1, 11], f(x)=$\frac{1}{11-1}$=$\frac{1}{10}$=0.1. Therefore, the correct choice is B, which corresponds to 0.1 as the probability density function for X.

Question 3

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 its properties of being symmetrical and bell-shaped, as well as its ability to approximate many other distributions. It is used widely in hypothesis testing, confidence intervals, and other statistical analyses.

Question 4

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, it is skewed to the right. The uniform and normal distributions are symmetrical. Therefore, the correct answer is A) Exponential.

Question 5

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

A normal distribution is unimodal (has one peak), symmetrical (mirror image on both sides), and has a bell shape (smooth, gradual curve). Therefore, all of these choices are correct for a normal distribution.

Question 6

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 deviati on of $X$.
D) None of these choices are correct.

Accepted Answer

For an exponentially distributed random variable, the mean is equal to the standard deviation.

Question 7

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 the sum of the areas from z = 0 to z = 1.30 and z = 0 to z = 1.50, which is 0.4032 + 0.4332 = 0.8364.

Question 8

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 rand om 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 means that the probability is distributed evenly across all possible values of X. Therefore, the height of the function (which represents the probability density) is the same for all values of X.

Question 9

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 .

Accepted Answer

The mean of an exponential distribution is 1/λ, and the mean of a Poisson distribution is λ. These means are equal when 1/λ = λ, which occurs when λ = 1. Thus, the mean of the exponential distribution is 1.0.

Question 10

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 of a normally distributed random variable is determined by its mean and variance. The mean specifies the location of the center of the distribution, while the variance controls the spread or width of the distribution. The other options are not sufficient to determine the probability density function of a normal random variable. For example, a normal distribution can have multiple modes or no mode at all, and the median may not coincide with the mean in skewed distributions.

Question 11

The probability density function, f(x), for any continuous random variable X, represents: 
A) all possible values that $X$ will assume within some interval $a \leq x \leq b$.
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 represents the relative likelihood of different values of the continuous random variable, but it does not give the probability that the random variable takes on a specific value. The area under the curve of the probability density function (PDF) within some interval represents the probability of X taking a value within that interval. The height of the function at a particular value of X gives some idea of the relative likelihood of X taking that value, but it does not represent a probability. Therefore, D is the best option as it represents the height of the function at a specific value of X.

Question 12

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 is the average of its minimum and maximum values which is $(a+b) / 2$.

Question 13

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

We know that for a normal distribution, approximately 2.5% of the values are  below µ-1.96σ. Therefore, µ-1.96σ=1.

Since the standard deviation is given as 2, we get µ-1.96(2)=1, which gives µ=4.92.

Question 14

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 uniform distribution is a constant value over the interval [a,b]. Since the total area under the probability density function must be equal to 1, we have:

∫f(x)dx = 1

Since the probability density function is constant over the interval [a,b], we can write:

∫f(x)dx = f(x)∫dx = f(x)(b-a)

Therefore,

f(x)(b-a) = 1

Solving for f(x), we get:

f(x) = 1 / (b-a)

So, the correct answer is A, which is 1/(b-a).

Question 15

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 ) = $\frac { 1 } { b - a } \text { if } a \leq x \leq b$

Accepted Answer

The standard deviation $\sigma$ for a uniform distribution is $(b-a)/\sqrt{12}$, not $(b-a)/6$.

Question 16

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.

Accepted Answer

The exponential distribution is appropriate for measuring the length of time between arrivals at a grocery checkout counter because it models the waiting time for an event to occur in a Poisson process, where events occur randomly and independently over time. In this case, customers arrive randomly and independently at the checkout counter, making the exponential distribution an appropriate choice for measuring the time between arrivals.

Question 17

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 density function of a uniform distribution is constant over the interval of support. In this case, the interval of support is [40, 60], so the probability density function is:```f(x) = 1 / (60 - 40) = 1 / 20 = 0.05```Therefore, the probability of X being between 35 and 45 is:```P(35 ≤ X ≤ 45) = ∫[35, 45] f(x) dx = ∫[35, 45] 0.05 dx = 0.05 * (45 - 35) = 0.25```

Question 18

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 distribution: 
z = (x - mu) / sigma 
where mu = mean, sigma = standard deviation. 
Rearranging the formula to solve for x: 
x = z * sigma + mu 
Substituting in the given values: 
x = 2.326 * 5 + 50 = 61.63 
Therefore, the z-value of 2.326 corresponds to the x-value of 61.63.

Question 19

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 )

Accepted Answer

The area to the left of a value z represents the probability that a standard normal random variable is less than or equal to z. This is expressed as P ( Z ≤ z).

Question 20

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

A continuous random variable does not have a finite number of possible outcomes like a binomial or Poisson distribution. Therefore, option D, the probability density function, is the best choice as it describes the density of the probability distribution over a continuous range of values. The normal distribution is a specific type of probability density function commonly used in statistics.

Quiz 8: Continuous Probability Distributions

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?

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?

Which of the following distributions is not symmetrical?

Which of the following is true for a normal distribution?

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

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

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

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

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

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

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?

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

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

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

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

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?

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

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

Quiz 8: Continuous Probability Distributions

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?

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?

Which of the following distributions is not symmetrical?

Which of the following is true for a normal distribution?

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

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

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

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

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

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

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?

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

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

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

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

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?

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

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

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

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

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