Question 1

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

Accepted Answer

Its mean is equal to 1.

Question 2

Given a uniform distribution on the interval $$[a, b]$$, what is the formula for the probability density function $$f(x)$$?

Accepted Answer

$$f(x) = \frac{1}{b-a}$$

Question 3

A waiting time at an airport security checkpoint is uniformly distributed between 0 and 5 minutes. What is the probability that a randomly selected passenger waits more than 2 minutes?

Accepted Answer

$$P(x \u003e 2) = \frac{5-2}{5-0} = 0.6$$

Question 4

Which formula correctly converts a value $$x$$ from a normal distribution with mean $$\mu$$ and standard deviation $$\sigma to a $$standard normal $$z$$ score?

Accepted Answer

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

Question 5

If the area to the left of $$z = 1.28$$ under the standard normal curve is approximately 0.8997, what is the probability that $$z \u003e 1.28$$?

Accepted Answer

$$1 - 0.8997 = 0.1003$$

Question 6

A bone density test result is normally distributed with mean 0 and standard deviation 1. What is the probability that a randomly selected adult has a result less than $$z = 1.27$$?

Accepted Answer

The area to the left of $$z = 1.27 in $$the standard normal table.

Question 7

Heights of men are normally distributed with mean $$68.6 in. $$and standard deviation $$2.8 in. $$What is the $$z$$ score for a man who is $$72$$ inches tall?

Accepted Answer

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

Question 8

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

Accepted Answer

As sample size increases, the distribution of sample means approaches a normal distribution, regardless of the population's distribution.

Question 9

If the mean of a population is $$\mu = 100$$ and the standard deviation is $$\sigma = 15$$, what is the standard error of the mean for samples of size $$n = 25$$?

Accepted Answer

$$\frac{15}{\sqrt{25}} = 3$$

Question 10

A sample mean $$\overline{x} is $$used to estimate the population mean $$\mu. $$What is $$\overline{x}$$ called?

Accepted Answer

An estimator

Question 11

Suppose the area to the left of $$z = a is 0.975. $$What is the value of $$a$$?

Accepted Answer

$$a \approx 1.96$$

Question 12

Which of the following is the 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 13

A sample of 36 has a mean of 50 and a population standard deviation of 12. What is the $$z$$ score for a sample mean of 53?

Accepted Answer

$$z = \frac{53 - 50}{12/\sqrt{36}} = \frac{3}{2} = 1.5$$

Question 14

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

Accepted Answer

The area under the curve for a given interval equals the probability that the random variable falls within that interval.

Question 15

If the distribution of a population is not normal and the sample size is less than or equal to 30, what can be said about the sampling distribution of the sample mean?

Accepted Answer

It cannot be well approximated by a normal distribution.

Question 16

A critical value $$z_{0.025} is $$defined as the $$z$$ score with what area to its right?

Accepted Answer

$$0.025$$

Question 17

Which of the following is a requirement for using the normal distribution to approximate the sampling distribution of the sample mean?

Accepted Answer

The population is normal or the sample size $$n \u003e 30.$$

Question 18

If the mean of a population is $$\mu$$ and the standard deviation is $$\sigma$$, what is the mean of the sampling distribution of the sample mean for samples of size $$n$$?

Accepted Answer

$$\mu$$

Normal Probability Distributions and the Central Limit Theorem: Study Guide

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