Q: A retail chain finds that 60% of its customers use debit cards. In a sample of 100 customers, what is the standard deviation of the sampling distribution of the sample proportion?

$$\sigma_{\bar{p}} = \sqrt{\frac{0.6 \times 0.4}{100}} = 0.049$$

Q: Suppose the IRS believes that 15% of all taxpayers will be affected by fraudulent tax returns. In a random sample of 200 taxpayers, what is the expected mean and standard deviation of the sample proportion?

Mean $$= 0.15$$, Standard deviation $$= \sqrt{\frac{0.15 \times 0.85}{200}} \approx 0.025$$

Q: Which formula correctly calculates the sampling error for a sample mean?

$$\text{Sampling Error} = \bar{x} - \mu$$

Question 1

Which of the following best defines sampling error?

Accepted Answer

The difference between the sample mean and the population mean due to random chance.

Question 2

Given a population with mean $$\mu = 71.43$$ and standard deviation $$\sigma = 16.80$$, what is the standard deviation of the sampling distribution of the sample mean for samples of size $$n = 25$$?

Accepted Answer

$$\sigma_{\bar{x}} = \frac{16.80}{\sqrt{25}} = 3.36$$

Question 3

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 approximately normal, regardless of the population's distribution.

Question 4

A food processing company produces French fries with a mean length of $$4.0$$ inches and a standard deviation of $$1.0$$ inch. What is the probability that a random sample of $$n = 100$$ fries has a mean length exceeding $$4.10$$ inches? (Hint: Find the $$z-$$score first.)

Accepted Answer

$$z = \frac{4.10 - 4.0}{1.0/\sqrt{100}} = 1.0$$

Question 5

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

Accepted Answer

It is equal to the population mean $$\mu.$$

Question 6

If the population is highly skewed, what is a conservative minimum sample size for the Central Limit Theorem to apply?

Accepted Answer

$$n \geq 30$$

Question 7

A retail chain finds that 60% of its customers use debit cards. In a sample of 100 customers, what is the standard deviation of the sampling distribution of the sample proportion?

Accepted Answer

$$\sigma_{\bar{p}} = \sqrt{\frac{0.6 	imes 0.4}{100}} = 0.049$$

Question 8

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

Accepted Answer

The standard deviation of the sampling distribution increases as sample size increases.

Question 9

Suppose the IRS believes that 15% of all taxpayers will be affected by fraudulent tax returns. In a random sample of 200 taxpayers, what is the expected mean and standard deviation of the sample proportion?

Accepted Answer

Mean $$= 0.15$$, Standard deviation $$= \sqrt{\frac{0.15 	imes 0.85}{200}} \approx 0.025$$

Question 10

A sample of size $$n = 10 is $$drawn from a population with mean $$\mu = 50$$ and standard deviation $$\sigma = 20. $$What is the probability that the sample mean is greater than $$55$$? (Hint: Find the $$z-$$score.)

Accepted Answer

$$z = \frac{55 - 50}{20/\sqrt{10}} = 1.58$$

Question 11

Which formula correctly calculates the sampling error for a sample mean?

Accepted Answer

$$	ext{Sampling Error} = \bar{x} - \mu$$

Question 12

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

Accepted Answer

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

Question 13

Which of the following best describes the Central Limit Theorem?

Accepted Answer

For large enough sample sizes, the sampling distribution of the sample mean is approximately normal, regardless of the population's distribution.

Question 14

A sample of 200 customers is taken from a population where the proportion using debit cards is $$p = 0.60. $$What is the $$z-$$score for observing a sample proportion $$\bar{p} = 0.48$$?

Accepted Answer

$$z = \frac{0.48 - 0.60}{\sqrt{\frac{0.6 	imes 0.4}{200}}} = -2.40$$

Question 15

Which of the following is a correct interpretation of a $$z-$$score in the context of sampling distributions?

Accepted Answer

It measures how many standard deviations a sample mean is from the population mean.

Question 16

If the sample size increases, what happens to the variability of the sample means?

Accepted Answer

The variability decreases.

Question 17

A population is not normally distributed. Which of the following statements is true 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 18

Which formula is used to calculate the standard deviation of the sampling distribution of the sample proportion?

Accepted Answer

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

Chapter 7: Introduction to Sampling Distributions – Business Statistics Study Notes

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