Question 1

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

Accepted Answer

The sampling distribution of the sample means becomes approximately normal as the sample size increases, regardless of the population's distribution.

Question 2

Suppose the population mean checkout time at a supermarket is 50.12 seconds with a standard deviation of 49.06 seconds. What is the standard deviation of the sampling distribution of the mean for samples of size $$n = 25$$?

Accepted Answer

$$\frac{49.06}{\sqrt{25}} = 9.812$$

Question 3

A packaging machine fills rice crispie bars with a mean weight of 22.2 g and a standard deviation of 0.40 g. What is the probability that a randomly selected bar weighs less than 22 g?

Accepted Answer

$$P(z \u003c \frac{22 - 22.2}{0.4}) = P(z \u003c -0.5) = 0.3085$$

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.

Question 5

If the true proportion of brown-haired people in a population is $$p = 0.55$$ and you take a random sample of $$n = 100$$, 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 6

Why does the sampling distribution of the sample mean have less variability than the population distribution?

Accepted Answer

Because the standard deviation of the sample mean is $$\frac{\sigma}{\sqrt{n}}$$, which decreases as $$n$$ increases.

Question 7

A researcher takes 100 samples of size $$n = 5$$ from a right-skewed population and calculates the mean of each sample. What will the histogram of these sample means most likely look like?

Accepted Answer

It will be more symmetric and closer to normal than the population distribution, but may still be slightly right-skewed.

Question 8

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 9

If you increase the sample size from $$n = 5 to$$ $$n = 25$$, what happens to the standard deviation of the sampling distribution of the mean?

Accepted Answer

It decreases by a factor of $$\sqrt{5}.$$

Question 10

A box contains 24 rice crispie bars. The mean weight of bars in the box is calculated. Which distribution should be used to find the probability that the mean weight is below a certain value?

Accepted Answer

The sampling distribution of the sample mean, modeled by the normal distribution using the Central Limit Theorem.

Question 11

Which of the following statements about sampling distributions is TRUE?

Accepted Answer

The mean of the sampling distribution of the sample mean is equal to the population mean.

Question 12

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 13

If a sample is not randomly selected, which assumption for the Central Limit Theorem is violated?

Accepted Answer

Randomization Condition

Question 14

A population is highly skewed to the right. What must be true for the sampling distribution of the sample mean to be approximately normal?

Accepted Answer

The sample size must be large enough.

Question 15

Which formula gives the standard deviation of the sampling distribution of the sample mean?

Accepted Answer

$$\sigma_{\bar{y}} = \frac{\sigma}{\sqrt{n}}$$

Question 16

Which formula gives the standard deviation of the sampling distribution of the sample proportion?

Accepted Answer

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

Chapter 14: Sampling Distribution Models – Study Notes

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