Q: Given $P(0 < z < 1.96)$ for the standard normal distribution, what does this probability represent?

The probability that $z$ is between 0 and 1.96.

Q: Suppose the heights of adult males are normally distributed with $\mu = 69.4$ inches and $\sigma = 2.4$ inches. What is the $z$-score for a height of 72 inches?

$z = \frac{72 - 69.4}{2.4} = 1.08$

Q: If $P(-1.26 \leq z \leq 1.26)$ is required, which method should be used to find this probability?

Subtract the area to the left of $z = -1.26$ from the area to the left of $z = 1.26$.

Q: Which real-world scenario is best modeled by a continuous probability distribution?

The height of students in a university class.

Q: What is the effect of increasing the standard deviation ($\sigma$) in a normal distribution while keeping the mean ($\mu$) constant?

The curve becomes wider and shorter.

Q: Which of the following is a valid use of the normal distribution in business statistics?

Estimating the probability of a specific sales amount occurring.

Q: A company finds that the weights of its product are normally distributed with $\mu = 500$g and $\sigma = 10$g. What proportion of products weigh more than 520g?

Find $z = \frac{520 - 500}{10} = 2$; look up $P(z > 2)$ in the standard normal table.

Question 1

Which of the following best describes a continuous random variable?

Accepted Answer

A variable that can take on any value within a given range.

Question 2

The probability density function (PDF) for a normal distribution is given by which formula?

Accepted Answer

$f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{1}{2} (\frac{x-\mu}{\sigma})^2}$

Question 3

What is the area under the entire normal curve?

Accepted Answer

$1$

Question 4

Which of the following statements about the normal distribution is FALSE?

Accepted Answer

It is always skewed to the right.

Question 5

If the mean ($\mu$) of a normal distribution increases while the standard deviation ($\sigma$) remains constant, what happens to the curve?

Accepted Answer

The curve shifts to the right.

Question 6

A standard normal random variable has which mean and standard deviation?

Accepted Answer

$\mu = 0$, $\sigma = 1$

Question 7

Given $P(0 < z < 1.96)$ for the standard normal distribution, what does this probability represent?

Accepted Answer

The probability that $z$ is between 0 and 1.96.

Question 8

Which of the following is NOT a property of the normal distribution?

Accepted Answer

The curve is always bimodal.

Question 9

Suppose the heights of adult males are normally distributed with $\mu = 69.4$ inches and $\sigma = 2.4$ inches. What is the $z$-score for a height of 72 inches?

Accepted Answer

$z = \frac{72 - 69.4}{2.4} = 1.08$

Question 10

If $P(-1.26 \leq z \leq 1.26)$ is required, which method should be used to find this probability?

Accepted Answer

Subtract the area to the left of $z = -1.26$ from the area to the left of $z = 1.26$.

Question 11

Which real-world scenario is best modeled by a continuous probability distribution?

Accepted Answer

The height of students in a university class.

Question 12

What is the effect of increasing the standard deviation ($\sigma$) in a normal distribution while keeping the mean ($\mu$) constant?

Accepted Answer

The curve becomes wider and shorter.

Question 13

Which of the following is a valid use of the normal distribution in business statistics?

Accepted Answer

Estimating the probability of a specific sales amount occurring.

Question 14

A company finds that the weights of its product are normally distributed with $\mu = 500$g and $\sigma = 10$g. What proportion of products weigh more than 520g?

Accepted Answer

Find $z = \frac{520 - 500}{10} = 2$; look up $P(z > 2)$ in the standard normal table.

Question 15

Which statement is TRUE about the relationship between discrete and continuous probability distributions?

Accepted Answer

Continuous distributions can be used to approximate some discrete distributions.

Question 16

If $P(z > 1.26)$ is required, which value should be used from the standard normal table?

Accepted Answer

The area to the right of $z = 1.26$.

Question 17

Which of the following is NOT a parameter of the normal distribution?

Accepted Answer

$\pi$ (pi)

Question 18

A random variable $X$ is said to be standard normal if:

Accepted Answer

$X$ has mean 0 and standard deviation 1.

Question 19

Which of the following best explains why the normal distribution is important in statistics?

Accepted Answer

It describes many random processes and is the basis for classical statistical inference.

Continuous Random Variables and the Standard Normal Distribution

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