Q: A pharmaceutical company wants to estimate the mean effect of a new drug. They collect a sample of 36 patients, with a sample mean effect of 8 units and a sample standard deviation of 3 units. What is the margin of error for a 90% confidence interval for the mean effect? (Use $$t_{0.05,35} \approx 1.690.)$$

$$ 1.690 \times \dfrac{3}{\sqrt{36}} = 0.845 $$

Question 1

Which of the following best describes a point estimate in statistics?

Accepted Answer

A value calculated from a sample that is used to estimate a population parameter.

Question 2

A researcher wants to estimate the proportion of defective items produced by a factory. She takes a random sample of 200 items and finds that 18 are defective. Construct a 95% confidence interval for the population proportion of defective items. Which formula should she use for the margin of error?

Accepted Answer

$$ E = z_{\alpha/2} \sqrt{ \dfrac{\hat{p}(1-\hat{p})}{n} } $$

Question 3

Suppose you want to estimate the mean systolic blood pressure of adults in a city. You collect a simple random sample of 25 adults, and the sample has a mean of 120 mmHg and a standard deviation of 15 mmHg. Which distribution should you use to construct a 99% confidence interval for the population mean?

Accepted Answer

Student's t-distribution with 24 degrees of freedom

Question 4

A sample of 16 batteries has a mean lifetime of 500 hours and a standard deviation of 40 hours. What is the correct formula for the 95% confidence interval for the population mean battery lifetime?

Accepted Answer

$$ \bar{x} \pm t_{\alpha/2} \dfrac{s}{\sqrt{n}} $$

Question 5

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

Accepted Answer

It represents the expected proportion of intervals that will contain the parameter if many samples are taken.

Question 6

A hospital administrator wants to estimate the standard deviation of patient wait times in the emergency room. She collects a random sample of 10 wait times and calculates a sample standard deviation of 12 minutes. Which distribution should she use to construct a confidence interval for the population standard deviation?

Accepted Answer

Chi-square distribution with 9 degrees of freedom

Question 7

Which of the following is the correct formula for a confidence interval for a population variance $$\$$sigma^2$$ based on a sample of size n$ with sample variance s^2$?

Accepted Answer

$$ \left( \dfrac{(n-1)s^2$}{\$$chi^2$$_{\alpha/2}}, \dfrac{(n-1)s^2$}{\$$chi^2$$_{1-\alpha/2}} \right) $$

Question 8

A pharmaceutical company wants to estimate the mean effect of a new drug. They collect a sample of 36 patients, with a sample mean effect of 8 units and a sample standard deviation of 3 units. What is the margin of error for a 90% confidence interval for the mean effect? (Use $$t_{0.05,35} \approx 1.690.)$$

Accepted Answer

$$ 1.690 	imes \dfrac{3}{\sqrt{36}} = 0.845 $$

Question 9

If a 95% confidence interval for a population mean is calculated as (45.5, 54.5), what is the correct interpretation?

Accepted Answer

We are 95% confident that the population mean lies between 45.5 and 54.5.

Question 10

A quality control manager wants to determine the minimum sample size needed to estimate the proportion of defective products within 2% of the true value with 95% confidence. If no prior estimate is available, which value should be used for $$\hat{p} in $$the sample size formula?

Accepted Answer

$$ 0.5 $$

Question 11

Which of the following is NOT a property of the Chi-Square distribution?

Accepted Answer

It is symmetric about its mean.

Question 12

A researcher is constructing a confidence interval for a population mean using a sample of size $$n=10. $$The population is known to be normally distributed. Which of the following is the correct number of degrees of freedom to use for the t-distribution?

Accepted Answer

$$ 9 $$

Question 13

A sample of 50 students has a mean test score of 78 and a standard deviation of 10. What is the 95% confidence interval for the mean test score if the population standard deviation is unknown? (Use $$t_{0.025,49} \approx 2.009.)$$

Accepted Answer

$$ 78 \pm 2.009 	imes \dfrac{10}{\sqrt{50}} $$

Question 14

Which of the following best explains why the t-distribution is used instead of the z-distribution when constructing confidence intervals for the mean with small samples?

Accepted Answer

Because the population standard deviation is unknown and the sample size is small.

Question 15

A biologist wants to estimate the variance of the weights of a species of fish. She collects a sample of 12 fish and finds a sample variance of 4.5 grams$$^2. $$What is the correct formula for the 99% confidence interval for the population variance?

Accepted Answer

$$ \left( \dfrac{11 	imes 4.5}{\$$chi^2$$_{0.005,11}}, \dfrac{11 	imes 4.5}{\$$chi^2$$_{0.995,11}} ight) $$

Question 16

A polling organization wants to estimate the proportion of voters who support a new policy. They want the estimate to be within 3% of the true proportion with 95% confidence. What is the correct formula for the minimum required sample size $$n if no $$prior estimate is available?

Accepted Answer

$$ n = \dfrac{z_{\alpha/2}^2 \cdot 0.25}{E^2$} $$

Question 17

A 99% confidence interval for a population mean is wider than a 90% confidence interval for the same data because:

Accepted Answer

A higher confidence level requires a larger margin of error.

Question 18

A medical researcher collects a sample of 30 patients and wants to estimate the mean recovery time after a new treatment. The sample mean is 14 days, and the sample standard deviation is 4 days. What is the 95% confidence interval for the mean recovery time? (Use $$t_{0.025,29} \approx 2.045.)$$

Accepted Answer

$$ 14 \pm 2.045 	imes \dfrac{4}{\sqrt{30}} $$

Question 19

Which of the following best describes the margin of error in a confidence interval?

Accepted Answer

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

Chapter 9: Confidence Intervals for Population Parameters

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