Q: Which of the following best describes the process of standardizing a normal random variable $X$?

Subtract the mean from $X$ and divide by the standard deviation: $Z = \frac{X - \mu}{\sigma}$

Q: A Z-score of -1.5 indicates that the value is:

1.5 standard deviations below the mean

Q: Which of the following is a real-world example of a continuous variable?

Temperature of a chemical solution

Q: A company wants to know the value of $X$ such that only 5% of items are thicker than $X$. If thickness is normally distributed with $\mu = 2.0$ mm and $\sigma = 0.1$ mm, and $P(Z > 1.645) = 0.05$, what is $X$?

$X = 2.0 + 1.645 \times 0.1 = 2.1645$ mm

Question 1

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

Accepted Answer

The mean, median, and mode are all equal.

Question 2

A continuous variable is best described as:

Accepted Answer

A variable that can assume any value on a continuum, limited only by measurement precision.

Question 3

Given a normal distribution with mean $\mu = 100$ and standard deviation $\sigma = 20$, what is the Z-score for $X = 140$?

Accepted Answer

$Z = \frac{140 - 100}{20} = 2$

Question 4

Which of the following statements about the standardized normal distribution (Z distribution) is TRUE?

Accepted Answer

It has a mean of 0 and a standard deviation of 1.

Question 5

Suppose the time to download a file is normally distributed with mean $\mu = 18.0$ seconds and standard deviation $\sigma = 5.0$ seconds. What is the probability that a randomly selected download takes less than 18.6 seconds? (Use the Z-table: $P(Z < 0.12) = 0.5478$)

Accepted Answer

0.5478

Question 6

For a normal distribution, what is the total area under the curve?

Accepted Answer

1.0

Question 7

If $P(Z < 0.12) = 0.5478$, what is $P(Z > 0.12)$?

Accepted Answer

0.4522

Question 8

Suppose $X$ is normally distributed with $\mu = 18.0$ and $\sigma = 5.0$. What is $P(18 < X < 18.6)$? (Given $P(Z < 0.12) = 0.5478$ and $P(Z < 0) = 0.5000$)

Accepted Answer

0.0478

Question 9

Which of the following best describes the process of standardizing a normal random variable $X$?

Accepted Answer

Subtract the mean from $X$ and divide by the standard deviation: $Z = \frac{X - \mu}{\sigma}$

Question 10

A company finds that the thickness of a manufactured item is normally distributed with mean $\mu = 2.0$ mm and standard deviation $\sigma = 0.1$ mm. What is the probability that a randomly selected item has a thickness greater than 2.15 mm? (Use $P(Z < 1.5) = 0.9332$)

Accepted Answer

0.0668

Question 11

If the mean, median, and mode of a distribution are all equal, and the distribution is symmetric and bell-shaped, which type of distribution is this?

Accepted Answer

Normal distribution

Question 12

Suppose the temperature of a solution is normally distributed with $\mu = 25^\circ$C and $\sigma = 2^\circ$C. What temperature corresponds to the 97.72th percentile? (Use $Z = 2.0$ for 97.72th percentile)

Accepted Answer

$25 + 2 	imes 2 = 29^\circ$C

Question 13

A Z-score of -1.5 indicates that the value is:

Accepted Answer

1.5 standard deviations below the mean

Question 14

If $P(Z < a) = 0.2000$, what is the value of $a$? (Use the Z-table: $P(Z < -0.84) = 0.2000$)

Accepted Answer

$a = -0.84$

Question 15

A hospital records the time to complete a certain medical test as normally distributed with mean $\mu = 30$ minutes and standard deviation $\sigma = 4$ minutes. What is the probability that a randomly selected test takes between 30 and 34 minutes? (Use $P(Z < 1) = 0.8413$ and $P(Z < 0) = 0.5000$)

Accepted Answer

0.3413

Question 16

Which of the following is a real-world example of a continuous variable?

Accepted Answer

Temperature of a chemical solution

Question 17

If the area to the left of $Z = 0.5$ is 0.6915, what is the probability that $Z$ is greater than 0.5?

Accepted Answer

0.3085

Question 18

A company wants to know the value of $X$ such that only 5% of items are thicker than $X$. If thickness is normally distributed with $\mu = 2.0$ mm and $\sigma = 0.1$ mm, and $P(Z > 1.645) = 0.05$, what is $X$?

Accepted Answer

$X = 2.0 + 1.645 	imes 0.1 = 2.1645$ mm

Question 19

Which of the following best explains why the probability of any single, exact value in a continuous distribution is zero?

Accepted Answer

Because the area under the curve at a single point is zero.

The Normal Distribution and Other Continuous Distributions

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