Q: A company wants to estimate the average customer satisfaction rating with a margin of error of 0.1 at 95% confidence. If the estimated standard deviation is 0.6, what is the minimum sample size required?

$$n = \left( \frac{1.96 \times 0.6}{0.1} \$$right)^2$$ = 138.3$$

Q: A sample of 25 bottles of juice has a mean fill volume of 492.21 mL and a standard deviation of 3.1 mL. What is the 99% confidence interval for the mean fill volume?

$$492.21 \pm 2.807 \times \frac{3.1}{\sqrt{25}}$$

Question 1

Which of the following best describes a point estimator?

Accepted Answer

A single statistic that estimates a population parameter.

Question 2

What is the formula for the standard error of a sample proportion $$\hat{p}$$?

Accepted Answer

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

Question 3

A researcher wants to estimate the proportion of smokers in a city with a margin of error of 0.03 at 95% confidence. If no prior estimate is available, what value should be used for $$p to $$obtain the most conservative sample size?

Accepted Answer

$$p = 0.5$$

Question 4

Which property is NOT required for a good estimator?

Accepted Answer

Large variance

Question 5

Suppose a sample of 200 individuals is tested for a gene mutation and 36 are found to be carriers. What is the point estimate $$\hat{p}$$ for the proportion of carriers?

Accepted Answer

$$\hat{p} = \frac{36}{200} = 0.18$$

Question 6

Which formula gives the margin of error for a confidence interval for a population mean when the population standard deviation is known?

Accepted Answer

$$z^* \frac{\sigma}{\sqrt{n}}$$

Question 7

Why do we use the $$t$$ distribution instead of the normal distribution when constructing a confidence interval for a mean with unknown population standard deviation?

Accepted Answer

The $$t$$ distribution accounts for extra uncertainty due to estimating the standard deviation from the sample.

Question 8

A company wants to estimate the average customer satisfaction rating with a margin of error of 0.1 at 95% confidence. If the estimated standard deviation is 0.6, what is the minimum sample size required?

Accepted Answer

$$n = \left( \frac{1.96 	imes 0.6}{0.1} \$$right)^2$$ = 138.3$$

Question 9

Which of the following statements about confidence intervals is TRUE?

Accepted Answer

A 95% confidence interval means that if we repeated the study many times, about 95% of the intervals would contain the true parameter.

Question 10

What is the formula for a 95% confidence interval for a population proportion $$p$$?

Accepted Answer

$$\hat{p} \pm 1.96 \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$$

Question 11

If the sample size increases, what happens to the margin of error for a confidence interval?

Accepted Answer

It decreases.

Question 12

Which of the following is an unbiased estimator for the population mean $$\mu$$?

Accepted Answer

The sample mean $$\bar{x}$$

Question 13

A sample of 25 bottles of juice has a mean fill volume of 492.21 mL and a standard deviation of 3.1 mL. What is the 99% confidence interval for the mean fill volume?

Accepted Answer

$$492.21 \pm 2.807 	imes \frac{3.1}{\sqrt{25}}$$

Question 14

Which of the following best describes the term 'margin of error'?

Accepted Answer

The maximum expected difference between the point estimate and the true parameter.

Question 15

When constructing a confidence interval for a mean using the $$t$$ distribution, what does the degrees of freedom ($$df) $$equal?

Accepted Answer

$$df = n - 1$$

Question 16

Which of the following is a reason interval estimates are preferred over point estimates?

Accepted Answer

Interval estimates provide a range of plausible values, reflecting uncertainty in the estimate.

Question 17

A pharmaceutical company wants to estimate the average recovery time (in days) for patients after a minor surgery. A sample of 7 patients has a mean recovery time of 4.2 days and a standard deviation of 0.95 days. What is the 95% confidence interval for the mean recovery time?

Accepted Answer

$$4.2 \pm 2.447 	imes \frac{0.95}{\sqrt{7}}$$

Estimation and Confidence Intervals: Structured Study Notes

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