Question 1

Which of the following statements about the area under a standard normal distribution curve is correct?

Accepted Answer

The area under the curve is equal to 1.

Question 2

Suppose a voltage is uniformly distributed between 10 and 16 volts. What is the probability that a randomly selected voltage is greater than 13.25 volts?

Accepted Answer

$$P = \frac{16 - 13.25}{16 - 10} = \frac{2.75}{6} \approx 0.458$$

Question 3

A standard normal distribution has which of the following properties?

Accepted Answer

Mean = 0, Standard deviation = 1

Question 4

Given a z-score, which of the following expressions correctly represents the probability that a value is less than $$z$$?

Accepted Answer

$$P(Z \u003c z)$$

Question 5

If the probability that a student scored between two z-values $$a$$ and $$b is $$needed, which formula should be used?

Accepted Answer

$$P(a \u003c Z \u003c b) = P(Z \u003c b) - P(Z \u003c a)$$

Question 6

A test score is normally distributed with a mean of 70 and a standard deviation of 8. What is the z-score for a student who scored 86?

Accepted Answer

$$z = \frac{86 - 70}{8} = 2$$

Question 7

Which of the following best describes the Central Limit Theorem?

Accepted Answer

The distribution of sample means approaches a normal distribution as the sample size increases.

Question 8

For a nonstandard normal distribution with mean $$\mu$$ and standard deviation $$\sigma$$, how do you convert a value $$x to a z-$$score?

Accepted Answer

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

Question 9

A population of heights is normally distributed with mean $$68$$ inches and standard deviation $$3$$ inches. What is the probability that a randomly selected person is taller than $$71$$ inches?

Accepted Answer

$$z = \frac{71 - 68}{3} = 1$$,\ P = 1 - P(Z \u003c 1) \approx 1 - 0.8413 = 0.1587$

Question 10

Which scenario requires using the sampling distribution of the sample mean $$\bar{x}$$ rather than the individual value distribution?

Accepted Answer

When you want the probability for the mean of a sample of size $$n.$$

Question 11

If the original population is not normally distributed, the sampling distribution of $$\bar{x} is $$approximately normal only when:

Accepted Answer

The sample size $$n \u003e 30$$

Question 12

Given a normal distribution with mean $$\mu = 100$$ and $$\sigma = 15$$, what is the value corresponding to the 90th percentile?

Accepted Answer

$$x = \mu + z_{0.90} \cdot \sigma = 100 + 1.28 \cdot 15 = 119.2$$

Question 13

A sample of 36 students has test scores that are normally distributed with mean $$70$$ and standard deviation $$12. $$What is the standard deviation of the sample mean?

Accepted Answer

$$\sigma_{\bar{x}} = \frac{12}{\sqrt{36}} = 2$$

Question 14

Which of the following is NOT a property of a uniform distribution?

Accepted Answer

The values are more likely to occur near the mean.

Question 15

If you are given a probability and asked to find the corresponding z-score, which tool or method would you use?

Accepted Answer

Use the inverse normal function or a z-table.

Question 16

A student scored in the 95th percentile on a normally distributed exam with mean $$80$$ and standard deviation $$10. $$What is the student's score?

Accepted Answer

$$x = 80 + 1.645 	imes 10 = 96.45$$

Question 17

Which of the following best describes a z-score?

Accepted Answer

The number of standard deviations a value is from the mean.

Question 18

Suppose exam scores are normally distributed with mean $$75$$ and standard deviation $$8. $$What is the probability that a student scored between $$70$$ and $$80$$?

Accepted Answer

$$z_1$$ = \frac{70 - 75}{8} = -0.625$$, z_2$ = \frac{80 - 75}{8} = 0.625$$; $$P = P(Z \u003c 0.625) - P(Z \u003c -0.625) \approx 0.734 - 0.266 = 0.468$$

Normal Probability Distributions and Sampling Distributions: Study Notes

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