Q: A bookstore wants to estimate the average length of books in its adult fiction section. The standard deviation is known to be 96 pages. What is the minimum sample size required to create a 99% confidence interval with a margin of error no more than 30 pages?

$$n = \left( \frac{2.576 \times 96}{30} \$$right)^2$$

Question 1

A researcher wants to estimate the average number of hours college students sleep per night. She takes a random sample of 36 students and finds a sample mean of 6.8 hours with a population standard deviation of 1.2 hours. Construct a 95% confidence interval for the population mean. Which formula should she use for the margin of error?

Accepted Answer

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

Question 2

Which of the following best describes the Central Limit Theorem (CLT)?

Accepted Answer

As the sample size $$n$$ increases, the sampling distribution of the sample mean $$\overline{X}$$ approaches a normal distribution, regardless of the population's distribution.

Question 3

A nutritionist wants to estimate the average grams of protein in a brand of protein bars. She takes a random sample of 40 bars with $$\overline{x} = 210 g $$and $$\sigma = 12 g. $$What is the 95% confidence interval for the mean protein content?

Accepted Answer

$$(210 - 3.72, 210 + 3.72)$$

Question 4

A bookstore wants to estimate the average length of books in its adult fiction section. The standard deviation is known to be 96 pages. What is the minimum sample size required to create a 99% confidence interval with a margin of error no more than 30 pages?

Accepted Answer

$$n = \left( \frac{2.576 	imes 96}{30} \$$right)^2$$

Question 5

A company’s marketing team takes 50 samples of 10 recent clients to create a sampling distribution of sample means for the average amount spent per month. They find that most of the 50 samples are right skewed. Can the Central Limit Theorem be used to determine that the sampling distribution is normal?

Accepted Answer

No, because the sample size is too small for the CLT to apply.

Question 6

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

Accepted Answer

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

Question 7

A sample of 20 packages has a variance of 4.5 lbs$$^2. $$Construct a 95% confidence interval for the population variance. Which distribution and degrees of freedom should be used?

Accepted Answer

Chi-square distribution with $$df = 19$$

Question 8

A previous study found that 80% of people preferred Pepsi over Coca Cola. Use a normal distribution to approximate the probability that at least 60 out of 100 people prefer Pepsi. What is the correct continuity correction for this calculation?

Accepted Answer

Use $$P(X \geq 59.5)$$

Question 9

Which of the following is NOT a requirement for using the normal approximation to the binomial distribution for sample proportions?

Accepted Answer

The population standard deviation is known

Question 10

A clinical researcher measures the systolic blood pressure (in mmHg) of 10 patients after taking a new medication. To construct a 95% confidence interval for the population variance, which table should be used?

Accepted Answer

Chi-square table

Question 11

A sample of 25 students spent an average of $$500 on textbooks with a sample standard deviation of 50. $$Assuming normality, which distribution should be used to construct a 98% confidence interval for the population mean?

Accepted Answer

t-distribution

Question 12

A puzzle company’s easiest puzzle is 100 pieces and the largest is 1000 pieces. Using the range rule of thumb, what is the estimated sample standard deviation?

Accepted Answer

$$s \approx \frac{1000 - 100}{4}$$

Question 13

Which of the following statements about confidence intervals for variance is correct?

Accepted Answer

The confidence interval for variance is not symmetric and does not use a margin of error.

Question 14

A researcher wants to estimate the proportion of Gen-Z that use social media with 90% confidence and a margin of error of 4%. What value of $$\hat{p}$$ should be used if no prior estimate is available?

Accepted Answer

$$\hat{p} = 0.5$$

Question 15

Which of the following is the correct formula for the degrees of freedom when using the t-distribution for a confidence interval for the mean?

Accepted Answer

$$df = n - 1$$

Question 16

A quality control specialist takes a sample of 15 parts and finds a sample variance of 2.5 mm$$^2. $$How does increasing the confidence level from 90% to 99% affect the width of the confidence interval for variance?

Accepted Answer

The interval becomes wider.

Sampling Distributions, Central Limit Theorem, and Confidence Intervals

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