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 becomes approximately normal, regardless of the population's distribution.

Question 2

Suppose the IQ scores in a population are normally distributed with mean $$\mu = 100$$ and standard deviation $$\sigma = 15. $$What is the standard error of the mean for samples of size $$n = 25$$?

Accepted Answer

$$\sigma_{\bar{x}} = \frac{15}{\sqrt{25}} = 3$$

Question 3

A population is highly skewed to the right. Which statement is true about the sampling distribution of the sample mean for samples of size $$n = 40$$?

Accepted Answer

The sampling distribution will be approximately normal due to the Central Limit Theorem.

Question 4

If the population standard deviation is $$\sigma = 12$$ and the sample size is $$n = 36$$, what is the standard error of the mean?

Accepted Answer

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

Question 5

A survey of 2540 musicians found that 368 are left-handed. What is the point estimate for the population proportion $$p of $$left-handed musicians?

Accepted Answer

$$\hat{p} = \frac{368}{2540} \approx 0.145$$

Question 6

According to the Central Limit Theorem, what is the minimum sample size $$n$$ generally recommended for the sampling distribution of the sample mean to be approximately normal, regardless of population shape?

Accepted Answer

$$n \geq 30$$

Question 7

A random sample of 40 oil changes has a mean time of 11.4 minutes and a standard deviation of 3.2 minutes. What is the probability that the sample mean is less than 10 minutes? (Assume normality by CLT.)

Accepted Answer

First, calculate the standard error: $$\sigma_{\bar{x}} = \frac{3.2}{\sqrt{40}} \approx 0.506. $$Then, $$z = \frac{10 - 11.4}{0.506} \approx -2.77. $$The probability is approximately 0.0028.

Question 8

If the population proportion is $$p = 0.26$$ and a random sample of $$n = 800 is $$taken, what is the standard deviation of the sampling distribution of $$\hat{p}$$?

Accepted Answer

$$\sigma_{\hat{p}} = \sqrt{\frac{0.26(1-0.26)}{800}} \approx 0.0156$$

Question 9

A computer repair shop has a mean daily revenue of $$\7200$ and a standard deviation of $$\$$1200. $$For the next 30 days, what is the probability that the mean daily revenue exceeds $$\7500$?

Accepted Answer

First, $$\sigma_{\bar{x}} = \frac{1200}{\sqrt{30}} \approx 219.09$$. $$z = \frac{7500 - 7200}{219.09} \approx 1.37$$. $$P(\bar{x} \u003e 7500) \approx 0.0855$$

Question 10

Which of the following is TRUE about the relationship between the population mean $$\mu$$ and the mean of the sample means $$\mu_{\bar{x}}$$?

Accepted Answer

$$\mu_{\bar{x}} = \mu$$

Question 11

A survey of 100 fatal accidents found that in 44 cases the driver at fault was inadequately insured. What is the sample proportion $$\hat{p}$$?

Accepted Answer

$$\hat{p} = \frac{44}{100} = 0.44$$

Question 12

Which formula gives the standard deviation of the sampling distribution of the sample proportion $$\hat{p}$$?

Accepted Answer

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

Question 13

If a population is normal with mean $$\mu = 50$$ and standard deviation $$\sigma = 10$$, what is the probability that a randomly selected sample of size $$n = 16$$ has a mean greater than 53?

Accepted Answer

First, $$\sigma_{\bar{x}} = \frac{10}{\sqrt{16}} = 2.5$$. $$z = \frac{53 - 50}{2.5} = 1.2$$. $$P(\bar{x} \u003e 53) \approx 0.1151$$

Question 14

Which condition must be met for the sampling distribution of $$\hat{p} to be $$approximately normal?

Accepted Answer

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

Question 15

A survey finds that 82% of Americans are satisfied with their lives. In a random sample of 100 Americans, what is the expected value (mean) of the sample proportion $$\hat{p}$$?

Accepted Answer

$$\mu_{\hat{p}} = 0.82$$

Question 16

Why do we use samples instead of populations in statistics?

Accepted Answer

It is often impractical or impossible to measure an entire population due to constraints on time, money, or effort.

Sampling Distributions: Mean and Proportion (Chapter 8 Study Guide)

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