Q: A business analyst wants to estimate the average time (in minutes) customers spend on a website. She collects a random sample of 36 customers and finds a sample mean of 12 minutes. The population standard deviation is known to be 3 minutes. What is the 95% confidence interval for the population mean?

$$12 \pm 1.96 \times \dfrac{3}{\sqrt{36}}$$

Q: A company wants to estimate the proportion of defective products. In a sample of 200 products, 18 are found to be defective. What is the standard error for the sample proportion?

$$\sqrt{\dfrac{0.09 \times 0.91}{200}}$$

Q: A survey of 1000 consumers finds that 26% value personalized experience the most when shopping. Construct a 90% confidence interval for the population proportion. What is the correct formula?

$$0.26 \pm 1.645 \times \sqrt{\dfrac{0.26 \times 0.74}{1000}}$$

Q: A random sample of 25 has a mean of 50 and a sample standard deviation of 8. What is the 95% confidence interval for the population mean? (Assume normality.)

$$50 \pm 2.064 \times \dfrac{8}{\sqrt{25}}$$

Q: A manager wants to estimate the mean processing time for insurance applications within 4 days with 90% confidence. A pilot study suggests the standard deviation is 25 days. What is the required sample size?

$$n = \left( \dfrac{1.645 \times 25}{4} \$$right)^2$$

Q: A random sample of 100 people shows that 25 are left-handed. What is the 95% confidence interval for the true proportion of left-handers?

$$0.25 \pm 1.96 \times \sqrt{\dfrac{0.25 \times 0.75}{100}}$$

Question 1

A business analyst wants to estimate the average time (in minutes) customers spend on a website. She collects a random sample of 36 customers and finds a sample mean of 12 minutes. The population standard deviation is known to be 3 minutes. What is the 95% confidence interval for the population mean?

Accepted Answer

$$12 \pm 1.96 	imes \dfrac{3}{\sqrt{36}}$$

Question 2

Which of the following statements best describes a 95% confidence interval for a population mean?

Accepted Answer

95% of all intervals constructed from repeated samples of the same size will contain the true mean.

Question 3

A company wants to estimate the proportion of defective products. In a sample of 200 products, 18 are found to be defective. What is the standard error for the sample proportion?

Accepted Answer

$$\sqrt{\dfrac{0.09 	imes 0.91}{200}}$$

Question 4

When is it appropriate to use the t-distribution instead of the normal (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 ($$n \leq 30).$$

Question 5

A survey of 1000 consumers finds that 26% value personalized experience the most when shopping. Construct a 90% confidence interval for the population proportion. What is the correct formula?

Accepted Answer

$$0.26 \pm 1.645 	imes \sqrt{\dfrac{0.26 	imes 0.74}{1000}}$$

Question 6

If the sample size increases while keeping the confidence level constant, what happens to the margin of error (MOE) in a confidence interval?

Accepted Answer

The margin of error decreases.

Question 7

A random sample of 25 has a mean of 50 and a sample standard deviation of 8. What is the 95% confidence interval for the population mean? (Assume normality.)

Accepted Answer

$$50 \pm 2.064 	imes \dfrac{8}{\sqrt{25}}$$

Question 8

Which of the following is NOT an assumption for constructing a confidence interval for the population mean when the population standard deviation is known?

Accepted Answer

The population mean is known.

Question 9

A manager wants to estimate the mean processing time for insurance applications within 4 days with 90% confidence. A pilot study suggests the standard deviation is 25 days. What is the required sample size?

Accepted Answer

$$n = \left( \dfrac{1.645 	imes 25}{4} \$$right)^2$$

Question 10

Which formula is used to determine the required sample size for estimating a population proportion with a specified margin of error $$e$$ and confidence level?

Accepted Answer

$$n = \dfrac{Z_{\alpha/2}^2 \; p(1-p)}{e^2$}$$

Question 11

Why do we use the sample standard deviation $$S$$ and the t-distribution when the population standard deviation $$\sigma is $$unknown?

Accepted Answer

Because using $$S$$ introduces extra uncertainty, and the t-distribution accounts for this variability.

Question 12

A random sample of 100 people shows that 25 are left-handed. What is the 95% confidence interval for the true proportion of left-handers?

Accepted Answer

$$0.25 \pm 1.96 	imes \sqrt{\dfrac{0.25 	imes 0.75}{100}}$$

Question 13

If you want to be more confident (e.g., increase the confidence level from 90% to 99%), what happens to the width of the confidence interval?

Accepted Answer

The interval becomes wider.

Question 14

Which of the following is the correct formula for the margin of error (MOE) in a confidence interval for the mean when $$\sigma is $$known?

Accepted Answer

$$MOE = Z_{\alpha/2} 	imes \dfrac{\sigma}{\sqrt{n}}$$

Question 15

A pilot sample yields $$p = 0.15. $$You want to estimate the true proportion within $$e = 0.07$$ with 95% confidence. What sample size is required?

Accepted Answer

$$n = \dfrac{(1.96)^2$ 	imes 0.15 	imes 0.85}{(0.07)^2$}$$

Question 16

Which of the following is TRUE about degrees of freedom in the context of the t-distribution for confidence intervals?

Accepted Answer

Degrees of freedom are calculated as $$n - 1$$, where $$n is $$the sample size.

Question 17

A marketing researcher wants to estimate the mean amount spent on Amazon by Prime members. A random sample of 101 members spent a mean of $$1500 with a standard deviation of 200. $$What is the 95% confidence interval for the mean amount spent?

Accepted Answer

$$1500 \pm 1.96 	imes \dfrac{200}{\sqrt{101}}$$

Question 18

Which of the following is a reason why confidence intervals are preferred over point estimates in business decision-making?

Accepted Answer

Confidence intervals provide a range of plausible values, reflecting uncertainty in the estimate.

Confidence Interval Estimation: Concepts, Formulas, and Applications

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