Q: Suppose the heights of women are normally distributed with $$\mu = 63.7 in. $$and $$\sigma = 2.9 in. $$What is the 95th percentile of heights for women?

$$x = 63.7 + 1.645 \times 2.9 = 68.5 in.$$

Question 1

Which of the following is the correct formula for the probability density function of a normal distribution with mean $$\mu$$ and standard deviation $$\sigma$$?

Accepted Answer

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

Question 2

A bone density test score is normally distributed with $$\mu = 0$$ and $$\sigma = 1. $$What is the probability that a randomly selected person has a score less than $$z = 1.27$$?

Accepted Answer

$$0.8389$$

Question 3

Suppose heights of men are normally distributed with $$\mu = 68.6 in. $$and $$\sigma = 2.8 in. $$What is the $$z-$$score for a man who is 72 inches tall?

Accepted Answer

$$z = \frac{72 - 68.6}{2.8} = 1.21$$

Question 4

If the area to the left of $$z = -1.00 is$$ $$0.1587$$, what is the probability that a randomly selected person has a $$z-$$score greater than $$-1.00$$?

Accepted Answer

$$0.8413$$

Question 5

A bone density reading between $$z = -1.00$$ and $$z = -2.50$$ indicates osteopenia. What is the probability that a randomly selected subject has a reading in this range?

Accepted Answer

$$0.1525$$

Question 6

Given a normal distribution with mean $$\mu$$ and standard deviation $$\sigma$$, what is the formula to convert an $$x$$ value to a $$z-$$score?

Accepted Answer

$$z = \frac{x - \mu}{\sigma}$$

Question 7

Suppose the heights of women are normally distributed with $$\mu = 63.7 in. $$and $$\sigma = 2.9 in. $$What is the 95th percentile of heights for women?

Accepted Answer

$$x = 63.7 + 1.645 	imes 2.9 = 68.5 in.$$

Question 8

Which of the following statements about the Central Limit Theorem is TRUE?

Accepted Answer

For any population distribution, the sampling distribution of the sample mean approaches a normal distribution as the sample size increases.

Question 9

A sample of $$n = 106$$ body temperatures has a mean of $$98.2^\circF. If $$the population mean is $$98.6^\circF $$with $$\sigma = 0.62^\circF$$, what is the $$z-$$score for the sample mean?

Accepted Answer

$$z = \frac{98.2 - 98.6}{0.62/\sqrt{106}}$$

Question 10

Which of the following is NOT a criterion for assessing normality of a data set?

Accepted Answer

The mean must be zero.

Question 11

If a sample statistic is an unbiased estimator, what does this mean?

Accepted Answer

Its sampling distribution has a mean equal to the population parameter.

Question 12

Why is sampling with replacement often used in probability theory?

Accepted Answer

It ensures that each selection is independent of previous selections.

Question 13

Which of the following is the correct interpretation of $$z_{0.025} in $$the context of the standard normal distribution?

Accepted Answer

It is the $$z-$$score with an area of $$0.025 to $$its right.

Question 14

A histogram of rainfall amounts is skewed to the right and the normal quantile plot shows points far from a straight line. What can you conclude?

Accepted Answer

The data are not from a normal distribution.

Question 15

If the area to the left of $$z = 1.96 is$$ $$0.975$$, what is the area to the right?

Accepted Answer

$$0.025$$

Question 16

Which of the following statistics is a biased estimator of its population parameter?

Accepted Answer

Sample range

Question 17

A sample of 27 adult males is selected. If the population mean weight is 189 lb and $$\sigma = 39 lb$$, what is the standard deviation of the sample mean?

Accepted Answer

$$\frac{39}{\sqrt{27}}$$

Question 18

Which of the following best describes the relationship between area under the normal curve and probability?

Accepted Answer

The area under the curve for a given interval equals the probability of a value falling within that interval.

Study Guide: Normal Probability Distributions in Statistics

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