Q: A researcher wants to estimate the mean systolic blood pressure of adult women in a city. She collects a random sample of 64 women and finds a sample mean of 120 mmHg. The population standard deviation is known to be 16 mmHg. Construct a 95% confidence interval for the population mean.

$$120 \pm 1.96 \times \frac{16}{\sqrt{64}}$$

Q: Which of the following is the correct formula for the margin of error $$E$$ when the population standard deviation $$\sigma is $$known?

$$E = z_{\alpha/2} \times \frac{\sigma}{\sqrt{n}}$$

Q: A company wants to estimate the average time it takes to assemble a product. They want the estimate to be within 2 minutes of the true mean with 95% confidence. The population standard deviation is estimated to be 6 minutes. What is the minimum sample size required?

$$n = \left( \frac{1.96 \times 6}{2} \$$right)^2$$

Q: A sample of 10 batteries has a mean life of 500 hours and a sample standard deviation of 20 hours. The population standard deviation is unknown. What is the 90% confidence interval for the mean battery life?

$$500 \pm t_{0.05,9} \times \frac{20}{\sqrt{10}}$$

Question 1

A researcher wants to estimate the mean systolic blood pressure of adult women in a city. She collects a random sample of 64 women and finds a sample mean of 120 mmHg. The population standard deviation is known to be 16 mmHg. Construct a 95% confidence interval for the population mean.

Accepted Answer

$$120 \pm 1.96 	imes \frac{16}{\sqrt{64}}$$

Question 2

Which of the following is the correct formula for the margin of error $$E$$ when the population standard deviation $$\sigma is $$known?

Accepted Answer

$$E = z_{\alpha/2} 	imes \frac{\sigma}{\sqrt{n}}$$

Question 3

A sample of 25 college students has a mean GPA of 3.2 with a sample standard deviation of 0.4. The population standard deviation is unknown. What distribution should be used to construct a confidence interval for the population mean?

Accepted Answer

Student's t-distribution with 24 degrees of freedom

Question 4

Which of the following best describes the point estimate for a population mean?

Accepted Answer

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

Question 5

A 95% confidence interval for the mean weight of a population is calculated as (170.5, 174.6) lbs. Which of the following statements is correct?

Accepted Answer

We are 95% confident that the population mean weight is between 170.5 and 174.6 lbs.

Question 6

If the sample size increases while keeping the confidence level and population standard deviation constant, what happens to the margin of error?

Accepted Answer

It decreases

Question 7

Which of the following is the correct formula for the degrees of freedom when using the t-distribution to estimate a population mean?

Accepted Answer

$$df = n - 1$$

Question 8

A company wants to estimate the average time it takes to assemble a product. They want the estimate to be within 2 minutes of the true mean with 95% confidence. The population standard deviation is estimated to be 6 minutes. What is the minimum sample size required?

Accepted Answer

$$n = \left( \frac{1.96 	imes 6}{2} \$$right)^2$$

Question 9

Which scenario requires the use of the standard normal (z) distribution rather than the t-distribution?

Accepted Answer

The population standard deviation $$\sigma is $$known

Question 10

A sample of 10 batteries has a mean life of 500 hours and a sample standard deviation of 20 hours. The population standard deviation is unknown. What is the 90% confidence interval for the mean battery life?

Accepted Answer

$$500 \pm t_{0.05,9} 	imes \frac{20}{\sqrt{10}}$$

Question 11

Which of the following best explains why the t-distribution is used instead of the normal distribution when $$\sigma is $$unknown?

Accepted Answer

The t-distribution accounts for the extra variability introduced by estimating $$\sigma$$ with $$s$$

Question 12

A 99% confidence interval is wider than a 95% confidence interval for the same data because:

Accepted Answer

A higher confidence level requires a larger critical value

Question 13

Which of the following is NOT a requirement for constructing a confidence interval for a population mean?

Accepted Answer

The population standard deviation must always be known

Question 14

A researcher calculates a 95% confidence interval for the mean height of a species of plant as (15.2, 18.6) cm. What does this interval mean?

Accepted Answer

We are 95% confident that the true mean height is between 15.2 and 18.6 cm

Question 15

If a sample mean is 50, the population standard deviation is 10, and the sample size is 25, what is the 90% confidence interval for the population mean?

Accepted Answer

$$50 \pm 1.645 	imes \frac{10}{\sqrt{25}}$$

Estimating Population Means and Confidence Intervals

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