Question 1

Which of the following best describes the Central Limit Theorem (CLT)?

Accepted Answer

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

Question 2

Suppose the population standard deviation is $$\sigma = 30$$ and you take a random sample of size $$n = 25. $$What is the standard deviation of the sampling distribution of the sample mean?

Accepted Answer

$$\$$\frac{30}{\sqrt{25}} = 6\$

Question 3

A company claims that the mean weight of its rice crispie bars is 22g with a standard deviation of 0.4g. If you randomly select a box of 24 bars, what is the standard deviation of the mean weight of the bars in the box?

Accepted Answer

$$\$$\frac{0.4}{\sqrt{24}} \approx 0.08165\$

Question 4

Which of the following is NOT an assumption required for the Central Limit Theorem to apply to sample means?

Accepted Answer

The population must be normally distributed, regardless of sample size.

Question 5

If a population is highly skewed, what must be true for the sampling distribution of the sample mean to be approximately normal?

Accepted Answer

The sample size must be large.

Question 6

Given a population proportion $$p = 0.55$$ and a sample size $$n = 100$$, what is the standard deviation of the sampling distribution of the sample proportion?

Accepted Answer

$$\$$\sqrt{\frac{0.55(1-0.55)}{100}} = 0.0497\$

Question 7

Which of the following conditions must be checked before using the normal model for the sampling distribution of sample proportions?

Accepted Answer

Randomization, 10% condition, and Success/Failure condition

Question 8

Why is the standard deviation of the sampling distribution of the sample mean smaller than the population standard deviation?

Accepted Answer

Because it is divided by the square root of the sample size: $$\$$\sigma_y = \frac{\sigma}{\sqrt{n}}\$

Question 9

A researcher takes 1000 samples of size 100 from a population with $$p = 0.55. $$What will the histogram of the sample proportions look like?

Accepted Answer

It will be approximately normal, centered at $$p = 0.55.$$

Question 10

Which of the following statements about sampling distributions is TRUE?

Accepted Answer

The mean of the sampling distribution of the sample mean equals the population mean.

Question 11

If you take a random sample of size $$n = 5$$ from a highly skewed population, what can you say about the sampling distribution of the sample mean?

Accepted Answer

It will still be somewhat skewed, not yet normal.

Question 12

What is the main purpose of using a sampling distribution in statistics?

Accepted Answer

To judge the reliability of a sample statistic as an estimator of a population parameter.

Question 13

Which of the following is a correct formula for the standard deviation of the sampling distribution of the sample proportion?

Accepted Answer

$$\$$\sqrt{\frac{p(1-p)}{n}}\$

Question 14

A sample of 100 rice crispie bars has a mean weight of 22.1g. The population mean is 22.2g and the population standard deviation is 0.4g. What is the z-score for the sample mean?

Accepted Answer

$$\z$$ = \frac{22.1 - 22.2}{0.4/\sqrt{100}} = \frac{-0.1}{0.04} = -2.5\$

Question 15

Which of the following is a common mistake when interpreting sampling distributions?

Accepted Answer

Confusing the sampling distribution with the distribution of the sample.

Question 16

Why is it important that sample observations are independent when applying the Central Limit Theorem?

Accepted Answer

Because dependence among observations can distort the sampling distribution, making the CLT invalid.

Chapter 14: Sampling Distribution Models – Central Limit Theorem and Applications

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