Q: Suppose you want to reduce the margin of error for a 95% confidence interval for the mean from 4.06 to 2, using the same sample standard deviation $$s = 9.84. $$What sample size should you target?

$$n = \left( \frac{1.96 \times 9.84}{2} \$$right)^2$$

Question 1

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

Accepted Answer

The sampling distribution of the mean becomes Normal as the sample size increases, regardless of the population's distribution.

Question 2

A company ships boxes with a mean weight of 12 lbs and a standard deviation of 4 lbs. If 10 boxes are shipped together, what is the standard deviation of the sampling distribution of the mean weight?

Accepted Answer

$$\frac{4}{\sqrt{10}}$$

Question 3

Which of the following is NOT an assumption or condition required for using the Student’s t-model for means?

Accepted Answer

The sample size must be greater than 100.

Question 4

A random sample of 25 customers has a mean age of 31.84 years and a standard deviation of 9.84 years. What is the standard error of the mean?

Accepted Answer

$$\frac{9.84}{\sqrt{25}}$$

Question 5

Which formula gives the confidence interval for the population mean $$\mu$$ when the population standard deviation is unknown?

Accepted Answer

$$\bar{y} \pm t^*_{n-1} \cdot SE(\bar{y})$$

Question 6

Why do we use $$n-1$$ instead of $$n in $$the denominator when calculating the sample standard deviation?

Accepted Answer

To compensate for estimating the population mean with the sample mean, which makes the sample standard deviation slightly too small if divided by $$n.$$

Question 7

A manager wants to test if the mean purchase amount is at least $$40. A random sample of 20 purchases has a mean of 45.26 $$and a standard deviation of $20.67. What is the correct null hypothesis?

Accepted Answer

$$H_0$$: \mu = 40$$

Question 8

Which of the following statements about the t-distribution is TRUE?

Accepted Answer

As the degrees of freedom increase, the t-distribution approaches the Normal distribution.

Question 9

Suppose you want to reduce the margin of error for a 95% confidence interval for the mean from 4.06 to 2, using the same sample standard deviation $$s = 9.84. $$What sample size should you target?

Accepted Answer

$$n = \left( \frac{1.96 	imes 9.84}{2} \$$right)^2$$

Question 10

Which of the following best explains why the sampling distribution of the mean has a smaller standard deviation than the population distribution?

Accepted Answer

Averaging over multiple observations reduces variability, so $$\sigma_{\bar{y}} = \frac{\sigma}{\sqrt{n}}.$$

Question 11

Which condition is most important to check before using the t-model for a sample of size 12?

Accepted Answer

The data should be nearly Normal (unimodal and symmetric) with no outliers.

Question 12

A histogram of CEO compensation is extremely skewed, but the histogram of sample means from samples of size 100 is nearly Normal. What concept does this illustrate?

Accepted Answer

The Central Limit Theorem

Question 13

Which formula is used to calculate the t-statistic for a one-sample t-test for the mean?

Accepted Answer

$$t = \frac{\bar{y} - \mu_0$}{SE(\bar{y})}$$

Question 14

If a 95% confidence interval for the mean age of customers is (27.78, 35.90), how should you interpret this interval?

Accepted Answer

We are 95% confident that the true mean age of all customers is between 27.78 and 35.90 years.

Question 15

Which of the following is a correct statement about the difference between the sampling distribution and the population distribution?

Accepted Answer

The sampling distribution describes the variability of a statistic (like the mean) from sample to sample, not the variability of individual data points.

Confidence Intervals and Hypothesis Tests for Means: Mini-Study Guide

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