Q: If $$X is $$normally distributed with mean $$\mu = 100$$ and standard deviation $$\sigma = 50$$, what is the value of $$X$$ that corresponds to $$Z = 1.5$$?

$$X = \mu + Z\sigma = 100 + 1.5 \times 50 = 175$$

Question 1

Which of the following statements best describes the shape and properties of a normal distribution?

Accepted Answer

It is symmetrical and bell-shaped, with the mean equal to the median.

Question 2

A university study finds that the daily commute time for students is normally distributed with a mean of 30 minutes and a standard deviation of 5 minutes. What is the Z-score for a student who commutes for 40 minutes?

Accepted Answer

$$Z = \frac{40 - 30}{5} = 2$$

Question 3

Which formula is used to convert a raw score $$X to a Z-$$score in a normal distribution?

Accepted Answer

$$Z = \frac{X - \mu}{\sigma}$$

Question 4

Suppose the time to download an image file from the internet is normally distributed with a mean of 18 seconds and a standard deviation of 5 seconds. What is the probability that a download takes less than 18.6 seconds?

Accepted Answer

Find $$Z = \frac{18.6 - 18}{5} = 0.12$$; use the cumulative standard normal table to get $$P(Z \u003c 0.12) \approx 0.5478$$

Question 5

If $$X is $$normally distributed with mean $$\mu = 100$$ and standard deviation $$\sigma = 50$$, what is the value of $$X$$ that corresponds to $$Z = 1.5$$?

Accepted Answer

$$X = \mu + Z\sigma = 100 + 1.5 	imes 50 = 175$$

Question 6

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

Accepted Answer

It is skewed to the right.

Question 7

A business analyst wants to know the probability that a randomly selected value from a normal distribution falls between $$a$$ and $$b. $$How is this probability represented?

Accepted Answer

As the area under the curve between $$a$$ and $$b$$: $$P(a \u003c X \u003c b)$$

Question 8

According to the empirical rule for a normal distribution, approximately what percentage of data falls within one standard deviation of the mean?

Accepted Answer

About 68%

Question 9

A company finds that the amount spent online before visiting a store is normally distributed with mean $$\mu = 250$$ and standard deviation $$\sigma = 20. $$What is the probability that a randomly selected amount is less than $$210$$?

Accepted Answer

Calculate $$Z = \frac{210 - 250}{20} = -2$$; $$P(Z \u003c -2) \approx 0.0228$$

Question 10

Which of the following best describes how to use a normal probability plot to evaluate normality?

Accepted Answer

If the plot is approximately linear, the data are approximately normally distributed.

Question 11

Suppose $$X is $$normal with mean $$18.0$$ and standard deviation $$5.0. $$What is the probability that $$X is $$greater than $$18.6$$?

Accepted Answer

Find $$Z = \frac{18.6 - 18}{5} = 0.12$$; $$P(X \u003e 18.6) = 1 - P(Z \u003c 0.12) \approx 1 - 0.5478 = 0.4522$$

Question 12

If 5% of the values in a normal distribution are less than $$X$$, what is the corresponding Z-score?

Accepted Answer

$$Z \approx -1.645$$

Question 13

Which of the following is true about the parameters $$\mu$$ and $$\sigma in $$the normal distribution?

Accepted Answer

$$\mu$$ determines the center and $$\sigma$$ determines the spread.

Question 14

A normal probability plot for mutual fund returns appears as a straight line. What does this indicate?

Accepted Answer

The data are approximately normally distributed.

Question 15

Given a normal distribution with mean $$\mu = 50$$ and standard deviation $$\sigma = 4$$, what is the probability that $$X \u003e 43$$?

Accepted Answer

Find $$Z = \frac{43 - 50}{4} = -1.75$$; $$P(X \u003e 43) = 1 - P(Z \u003c -1.75) \approx 1 - 0.0401 = 0.9599$$

Question 16

Which of the following is a continuous random variable?

Accepted Answer

Time required to complete a task

Question 17

If the interquartile range of a normal distribution is approximately 1.33 standard deviations, what does this tell you about the spread of the middle 50% of the data?

Accepted Answer

The middle 50% of the data lies within $$\pm 0.665$$ standard deviations from the mean.

Question 18

A normal probability plot for a data set shows a pronounced curve upward. What does this indicate about the data?

Accepted Answer

The data are right-skewed and deviate from normality.

Question 19

Suppose $$X is $$normal with mean $$18.0$$ and standard deviation $$5.0. $$What is the probability that $$17.4 \u003c X \u003c 18$$?

Accepted Answer

Find $$Z_1$$ = \frac{17.4 - 18}{5} = -0.12$$, Z_2$ = \frac{18 - 18}{5} = 0$$; $$P(-0.12 \u003c Z \u003c 0) = P(Z \u003c 0) - P(Z \u003c -0.12) \approx 0.5 - 0.4522 = 0.0478$$

Normal Distribution: Concepts, Applications, and Evaluating Normality

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