Q: If the population proportion is $$p = 0.8$$ and a sample of $$n = 20 is $$taken, can the normal approximation be used for the sampling distribution of $$\hat{p}$$?

No, because $$n(1-p) = 20 \times 0.2 = 4 \u003c 15.$$

Question 1

Which of the following best describes a sampling distribution?

Accepted Answer

A distribution of all possible values of a statistic computed from samples of the same size from a population.

Question 2

According to the Central Limit Theorem, what happens to the shape of the sampling distribution of the sample mean as the sample size increases?

Accepted Answer

It becomes more normal, regardless of the population's distribution.

Question 3

Given a population with mean $$\mu = 50$$ and standard deviation $$\sigma = 10$$, what is the standard deviation of the sampling distribution of the sample mean for samples of size $$n = 25$$?

Accepted Answer

$$\sigma_{\bar{x}} = \frac{10}{\sqrt{25}} = 2$$

Question 4

A population has a mean $$\mu = 100$$ and standard deviation $$\sigma = 20. If $$you take a random sample of $$n = 64$$, what is the probability that the sample mean is greater than 105? (Assume normality.)

Accepted Answer

First, calculate $$\sigma_{\bar{x}} = \frac{20}{\sqrt{64}} = 2.5. $$Then, $$z = \frac{105 - 100}{2.5} = 2. $$The probability is $$P(z \u003e 2) = 0.0228.$$

Question 5

Which condition must be met to use the normal approximation for the sampling distribution of the sample proportion $$\hat{p}$$?

Accepted Answer

$$np \u003e 15$$ and $$n(1-p) \u003e 15$$

Question 6

If the population proportion is $$p = 0.8$$ and a sample of $$n = 20 is $$taken, can the normal approximation be used for the sampling distribution of $$\hat{p}$$?

Accepted Answer

No, because $$n(1-p) = 20 	imes 0.2 = 4 \u003c 15.$$

Question 7

A sample of 200 students is surveyed, and 80% prefer Coke. What is the mean and standard deviation of the sampling distribution of $$\hat{p}$$?

Accepted Answer

Mean: $$\mu_{\hat{p}} = 0.8$$, Standard deviation: $$\sigma_{\hat{p}} = \sqrt{\frac{0.8 	imes 0.2}{200}} = 0.0283$$

Question 8

Suppose the mean body temperature in a population is $$98.6^{\circ}F $$with $$\sigma = 0.62$$^{\circ}$$F. For a sample of n = 106$, what is the probability of getting a sample mean of 98.2$$^{\circ}$$F or lower?

Accepted Answer

Calculate $$\sigma_{\bar{x}} = \frac{0.62}{\sqrt{106}} = 0.060$$. $$z = \frac{98.2 - 98.6}{0.060} = -6.67$$. $$P(z \u003c -6.67) \approx 0.$$

Question 9

A wireless phone carrier reports $$\mu = 3.7$$ minutes and $$\sigma = 1.9$$ minutes for call duration. For a sample of $$n = 40$$ calls, what is the standard deviation of the sampling distribution of the sample mean?

Accepted Answer

$$\sigma_{\bar{x}} = \frac{1.9}{\sqrt{40}} = 0.3004$$

Question 10

If a sample mean is 4.4 minutes, population mean is 3.7 minutes, and $$\sigma_{\bar{x}} = 0.3$$, what is the z-score for the sample mean?

Accepted Answer

$$z = \frac{4.4 - 3.7}{0.3} = 2.33$$

Question 11

A mayoral candidate claims 53% support. In a sample of 400, what is the probability of getting a sample proportion less than 49%?

Accepted Answer

Calculate $$\sigma_{\hat{p}} = \sqrt{\frac{0.53 	imes 0.47}{400}} = 0.025$$. $$z = \frac{0.49 - 0.53}{0.025} = -1.6$$. $$P(z \u003c -1.6) = 0.0548.$$

Question 12

Which of the following statements is TRUE about the spread of the sampling distribution of the sample mean?

Accepted Answer

The spread decreases as the sample size increases.

Question 13

If the population distribution is highly skewed, what can you say about the sampling distribution of the sample mean for large $$n$$?

Accepted Answer

It will be approximately normal due to the Central Limit Theorem.

Question 14

A sample of 1,252 debit cardholders found 180 used their card online. If the true population proportion is 15%, what is the sample proportion $$\hat{p}$$?

Accepted Answer

$$\hat{p} = \frac{180}{1252} = 0.144$$

Question 15

Which of the following is NOT a property of the sampling distribution of the sample mean?

Accepted Answer

It is always skewed, regardless of sample size.

Question 16

If $$n = 1000$$ and $$p = 0.53$$, what is the standard deviation of the sampling distribution of $$\hat{p}$$?

Accepted Answer

$$\sigma_{\hat{p}} = \sqrt{\frac{0.53 	imes 0.47}{1000}} = 0.015$$

Sampling Distributions for Means and Proportions: Central Limit Theorem and Applications

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