Q: A random sample of 100 voters shows that 40 support a new policy. What is the 95% confidence interval for the true proportion of supporters?

$$0.40 \pm 1.96 \sqrt{\frac{0.4 \times 0.6}{100}}$$

Q: A company wants to estimate the proportion of defective products with a margin of error of 2% and 95% confidence. If a pilot study suggests $$p = 0.10$$, what is the minimum sample size required? (Use $$Z_{0.025} = 1.96)$$

$$n = \frac{(1.96)^2$ \times 0.10 \times 0.90}{0.02^2$}$$

Question 1

Which of the following best describes a confidence interval?

Accepted Answer

A range of values, derived from a sample, that is likely to contain the population parameter with a specified level of confidence.

Question 2

A researcher wants to estimate the average time (in minutes) it takes for a new medication to take effect. The population standard deviation is known to be 12 minutes. What is the formula for the 95% confidence interval for the mean if a random sample of 36 patients has a mean of 50 minutes?

Accepted Answer

$$50 \pm 1.96 \frac{12}{\sqrt{36}}$$

Question 3

Which of the following statements about the confidence level is TRUE?

Accepted Answer

A 95% confidence level means that 95% of all possible confidence intervals constructed from repeated samples will contain the true population parameter.

Question 4

When should you use the t-distribution instead of the z-distribution for constructing a confidence interval for the mean?

Accepted Answer

When the population standard deviation is unknown and the sample size is small (n \u003c 30).

Question 5

A sample of 25 employees has a mean salary of $$60,000 and a sample standard deviation of 8$$,000. What is the 95% confidence interval for the population mean salary? (Use $$t_{0.025,24} = 2.064)$$

Accepted Answer

$$60,000 \pm 2.064 \frac{8,000}{\sqrt{25}}$$

Question 6

Which of the following will NOT decrease the width of a confidence interval for the mean?

Accepted Answer

Increasing the population standard deviation.

Question 7

A random sample of 100 voters shows that 40 support a new policy. What is the 95% confidence interval for the true proportion of supporters?

Accepted Answer

$$0.40 \pm 1.96 \sqrt{\frac{0.4 	imes 0.6}{100}}$$

Question 8

Which of the following is a correct interpretation of a 99% confidence interval for a population mean?

Accepted Answer

If we were to take many samples and build a confidence interval from each, about 99% of those intervals would contain the true population mean.

Question 9

A company wants to estimate the proportion of defective products with a margin of error of 2% and 95% confidence. If a pilot study suggests $$p = 0.10$$, what is the minimum sample size required? (Use $$Z_{0.025} = 1.96)$$

Accepted Answer

$$n = \frac{(1.96)^2$ 	imes 0.10 	imes 0.90}{0.02^2$}$$

Question 10

Which of the following is an assumption for constructing a confidence interval for the mean when the population standard deviation is known?

Accepted Answer

The population must be normally distributed, or the sample size must be large (n \u003e 30).

Question 11

A marketing firm wants to compare the average ratings of an advertisement between boys and girls using paired samples. Which formula should be used to construct a confidence interval for the mean difference?

Accepted Answer

$$\bar{d} \pm t_{\alpha/2, n-1} \frac{S_d}{\sqrt{n}}$$

Question 12

If the sample size increases while keeping all other factors constant, what happens to the width of the confidence interval for the mean?

Accepted Answer

It decreases.

Question 13

Which of the following is NOT an ethical guideline when reporting confidence interval estimates?

Accepted Answer

Only report the interval if it supports your hypothesis.

Question 14

A sample of 80 IT managers has a mean Emotional Quotient (EQ) of 93.06 and a standard deviation of 10.42. What is the standard error of the mean?

Accepted Answer

$$\frac{10.42}{\sqrt{80}}$$

Question 15

Which of the following is TRUE about the t-distribution compared to the normal (z) distribution?

Accepted Answer

The t-distribution has fatter tails than the normal distribution, especially for small sample sizes.

Question 16

A news organization wants to estimate the proportion of people who believe in global warming to within 1% with 99% confidence. What critical value should be used in the sample size formula?

Accepted Answer

$$Z_{0.005} = 2.576$$

Question 17

Which of the following is a correct formula for the margin of error in a confidence interval for a population proportion?

Accepted Answer

$$Z_{\alpha/2} \sqrt{\frac{p(1-p)}{n}}$$

Confidence Interval Estimation – Business Statistics Chapter 8 Study Notes

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