Question 1

Which of the following statements about the normal density curve is TRUE?

Accepted Answer

It is symmetric about its mean $$\mu$$ and has a single peak at $$x = \mu.$$

Question 2

A pediatrician measures the heights of three-year-old female patients, which are approximately normally distributed with mean $$\mu = 37.2$$ inches and standard deviation $$\sigma = 3.17$$ inches. What is the $$z-$$score for a child who is 36.0 inches tall?

Accepted Answer

$$z = \dfrac{36.0 - 37.2}{3.17} = -1.17$$

Question 3

According to the Empirical Rule, approximately what percentage of data falls within two standard deviations of the mean in a normal distribution?

Accepted Answer

95%

Question 4

In the context of the Central Limit Theorem, which formula gives the standard deviation of the sampling distribution of the sample mean for samples of size $$n$$?

Accepted Answer

$$\dfrac{\sigma}{\sqrt{n}}$$

Question 5

A binomial random variable $$X$$ with parameters $$n$$ and $$p$$ can be approximated by a normal distribution when which of the following conditions is met?

Accepted Answer

$$np \u003e 10$$ and $$n(1-p) \u003e 10$$

Question 6

In a sample of 500 people, 7% are expected to have blood type O-negative. What is the mean and standard deviation for the number of people with blood type O-negative?

Accepted Answer

$$\mu = 35$$, $$\sigma = 5.71$$

Question 7

If the $$z-$$score for a value $$x in a $$normal distribution is negative, what does this indicate about the value of $$x$$?

Accepted Answer

It is less than the mean.

Question 8

A Boeing 737 seat is 16.0 inches wide. The hip widths of adult males are normally distributed with mean $$\mu = 14.3$$ inches and standard deviation $$\sigma = 1.0$$ inch. What is the $$z-$$score for a hip width of 16.0 inches?

Accepted Answer

$$z = \dfrac{16.0 - 14.3}{1.0} = 1.7$$

Question 9

Which property of the normal density curve allows us to use the area under the curve to the left of $$x to $$find probabilities?

Accepted Answer

The area under the curve to the left of $$x$$ equals the probability that a randomly selected value is less than $$x.$$

Question 10

A sample of 126 adult males is taken. The mean hip width is $$\mu = 14.3$$ inches and the standard deviation of the sample mean is $$\sigma_{\bar{x}} = 0.089. $$What is the probability that the sample mean hip width is greater than 16.0 inches?

Accepted Answer

Approximately 0.0001

Question 11

Which of the following is NOT a property of the normal density curve?

Accepted Answer

It is skewed to the right.

Question 12

If the mean of a normal distribution is $$\mu = 35$$ and the standard deviation is $$\sigma = 5.71$$, what is the $$z-$$score for $$x = 29.5$$?

Accepted Answer

$$z = \dfrac{29.5 - 35}{5.71} = -0.96$$

Question 13

Which statement best describes the application of the Central Limit Theorem?

Accepted Answer

It allows us to approximate the distribution of sample means as normal when the sample size is large.

Question 14

A normal curve has mean $$\mu = 0$$ and standard deviation $$\sigma = 1. $$What is the probability that a randomly selected value falls between $$\mu - 2\sigma$$ and $$\mu + 2\sigma$$?

Accepted Answer

Approximately 95%

Question 15

In a binomial experiment with $$n = 500$$ and $$p = 0.07$$, what is the formula for the standard deviation of the number of successes?

Accepted Answer

$$\sigma = \sqrt{np(1-p)}$$

Question 16

Why is the normal approximation to the binomial distribution appropriate in the blood type O-negative example with $$n = 500$$ and $$p = 0.07$$?

Accepted Answer

Because both $$np$$ and $$n(1-p)$$ are greater than 10.

Question 17

A pediatrician wants to find the probability that a three-year-old female patient is shorter than 36.0 inches. The heights are normally distributed with $$\mu = 37.2$$ and $$\sigma = 3.17. $$Which steps should be followed?

Accepted Answer

Convert 36.0 to a $$z-$$score, then use the normal table to find the area to the left.

Question 18

If the area under the standard normal curve to the left of $$z = -0.96 is 0.1685$$, what does this represent in a real-world context?

Accepted Answer

The probability that a randomly selected value is less than $$z = -0.96.$$

Normal Distribution, Central Limit Theorem, and Applications in Statistics

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