Q: A market researcher surveys 100 customers and finds a sample mean expenditure of $$32.51 with a sample standard deviation of 25.7. $$What is the 95% confidence interval for the population mean?

$$32.51 \pm 2 \times \frac{25.7}{\sqrt{100}}$$

Q: A poll of 100 voters finds that 52 would vote for Candidate X. What is the standard error for the sample fraction?

$$\sqrt{0.52 \times (1-0.52) / 100}$$

Q: A national poll wants to estimate the fraction of voters who will vote for a candidate with a margin of error of 2.5% at 95% confidence. Assuming the worst-case scenario for the standard deviation, what sample size should be used?

$$n = 16 \times 0.5^2$ / 0.025^2$

Question 1

Which of the following best describes the concept of 'sampling error' in statistical inference?

Accepted Answer

The error that occurs when the sample mean is different from the population mean due to random chance.

Question 2

A market researcher surveys 100 customers and finds a sample mean expenditure of $$32.51 with a sample standard deviation of 25.7. $$What is the 95% confidence interval for the population mean?

Accepted Answer

$$32.51 \pm 2 	imes \frac{25.7}{\sqrt{100}}$$

Question 3

Which of the following statements about sample size and confidence intervals is TRUE?

Accepted Answer

Increasing the sample size decreases the length of the confidence interval.

Question 4

Suppose you want a 95% confidence interval for a population mean to have a total length $$L. $$What sample size $$n$$ should you use?

Accepted Answer

$$n = \frac{16s^2$}{L^2$}$$

Question 5

A poll of 100 voters finds that 52 would vote for Candidate X. What is the standard error for the sample fraction?

Accepted Answer

$$\sqrt{0.52 	imes (1-0.52) / 100}$$

Question 6

Which of the following is a correct interpretation of a 95% confidence interval for a population mean?

Accepted Answer

If we took many samples, about 95% of the confidence intervals would contain the true population mean.

Question 7

A sample mean is $$m = 32.51$$, the sample standard deviation is $$s = 25.7$$, and the sample size is $$n = 100. $$What is the value of the t-statistic for testing if the population mean is greater than a critical value $$c = 27$$?

Accepted Answer

$$t = \frac{32.51 - 27}{25.7 / \sqrt{100}}$$

Question 8

Which of the following is NOT a source of error in making inferences from a sample?

Accepted Answer

Using a sufficiently large random sample.

Question 9

Why is the divisor $$n-1$$ used instead of $$n$$ when calculating the sample standard deviation?

Accepted Answer

To make the sample standard deviation an unbiased estimator of the population standard deviation.

Question 10

A national poll wants to estimate the fraction of voters who will vote for a candidate with a margin of error of 2.5% at 95% confidence. Assuming the worst-case scenario for the standard deviation, what sample size should be used?

Accepted Answer

$$n = 16 	imes 0.5^2$ / 0.025^2$

Question 11

Which of the following best describes 'response bias' in survey sampling?

Accepted Answer

Systematic error caused by the way questions are asked or answered.

Question 12

If a sample is not truly random, which of the following is a potential consequence?

Accepted Answer

The sample may not be representative, leading to biased inferences.

Question 13

Which formula gives the standard error of the sample mean?

Accepted Answer

$$\frac{s}{\sqrt{n}}$$

Question 14

A sample of 1000 people is drawn from a population of 1,000,000. Ignoring the finite population correction (fpc) in this case will:

Accepted Answer

Have negligible effect on the standard error.

Question 15

Which of the following is a correct shortcut formula for the sample standard deviation $$s of a $$dummy variable (coded 0 or 1) with sample fraction $$f$$?

Accepted Answer

$$s = \sqrt{f(1-f)}$$

Question 16

Which of the following is TRUE about the relationship between statistical significance and practical importance?

Accepted Answer

A result can be statistically significant but not practically important.

Question 17

If you quadruple the sample size $$n$$, what happens to the length of the confidence interval for the mean?

Accepted Answer

It is halved.

Sampling and Statistical Inference: Core Concepts for Introductory Statistics

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