Question 1

Which of the following statements about the normal distribution is TRUE?

Accepted Answer

The normal distribution is symmetric about its mean, and the total area under the curve is 1.

Question 2

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 3

A researcher collects a sample of size $$n = 50$$ from a population with mean $$\mu = 100$$ and standard deviation $$\sigma = 15. $$What is the standard deviation of the sampling distribution of the sample mean?

Accepted Answer

$$\frac{15}{\sqrt{50}}$$

Question 4

Which formula is used to standardize a value $$x$$ from a normal distribution with mean $$\mu$$ and standard deviation $$\sigma$$?

Accepted Answer

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

Question 5

If a population is not normally distributed, under what condition will the sampling distribution of the sample mean be approximately normal?

Accepted Answer

If the sample size is at least 30, according to the Central Limit Theorem.

Question 6

A sample proportion $$\hat{p} is $$considered approximately normal if which condition is met?

Accepted Answer

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

Question 7

Which of the following best describes the difference between a parameter and a statistic?

Accepted Answer

A parameter is a fixed value describing a population, while a statistic is a variable describing a sample.

Question 8

Suppose the mean of a population is $$\mu = 80$$ and the standard deviation is $$\sigma = 12. $$What is the $$z-$$score for a value $$x = 104$$?

Accepted Answer

$$z = 2$$

Question 9

Why do we use the standard normal distribution (z-table) instead of calculating probabilities for every possible mean and standard deviation?

Accepted Answer

Because standardizing allows us to use a single table for all normal distributions.

Question 10

A sample of 100 students has a mean test score of 75. The population standard deviation is 10. What is the probability that a randomly selected sample of 100 students will have a mean score greater than 77? (Assume normality and use the standard normal distribution.)

Accepted Answer

Find $$z = \frac{77-75}{\frac{10}{\sqrt{100}}} = 2$$, so $$P(\bar{x} \u003e 77) = 1 - 0.9772 = 0.0228$$

Question 11

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

Accepted Answer

The area under the curve can be greater than 1.

Question 12

If the mean of the sampling distribution of $$\hat{p} is $$equal to the population proportion $$p$$, what does this imply?

Accepted Answer

$$\hat{p} is an $$unbiased estimator of $$p.$$

Question 13

A population has a mean $$\mu = 50$$ and standard deviation $$\sigma = 8. If $$you take a random sample of size $$n = 64$$, what is the standard deviation of the sample mean?

Accepted Answer

$$\frac{8}{\sqrt{64}} = 1$$

Question 14

Which of the following best describes the Central Limit Theorem?

Accepted Answer

The sampling distribution of the sample mean will be approximately normal if the sample size is large enough, regardless of the population's shape.

Question 15

Which notation correctly represents the mean of the sampling distribution of the sample mean?

Accepted Answer

$$\mu_{\bar{x}}$$

Question 16

A sample of size $$n = 40 is $$drawn from a population with $$p = 0.3. $$What is the standard deviation of the sampling distribution of $$\hat{p}$$?

Accepted Answer

$$\sqrt{\frac{0.3(1-0.3)}{40}}$$

Question 17

Why does increasing the sample size reduce the variability of the sampling distribution?

Accepted Answer

Because the standard deviation of the sampling distribution decreases as $$n$$ increases.

Question 18

Which of the following is a correct interpretation of the area under a normal density curve between two values $$a$$ and $$b$$?

Accepted Answer

It represents the proportion of data values between $$a$$ and $$b.$$

Question 19

A company claims that the average lifetime of its light bulbs is 1200 hours with a standard deviation of 100 hours. If a quality control manager takes a random sample of 36 bulbs, what is the probability that the sample mean lifetime is less than 1170 hours? (Assume normality.)

Accepted Answer

Find $$z = \frac{1170-1200}{\frac{100}{\sqrt{36}}} = -1.8$$, so $$P(\bar{x} \u003c 1170) = 0.0359$$

Normal and Sampling Distributions: Study Guide (Chapters 7 & 8)

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