Q: A sample of 30 songs has a mean length of 250 seconds and a sample standard deviation of 85 seconds. What is the 95% confidence interval for the mean song length? (Use $$t^* = 2.042$$ for 29 degrees of freedom.)

$$250 \pm 2.042 \times \frac{85}{\sqrt{30}}$$

Question 1

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

Accepted Answer

The sampling distribution of the sample mean will be approximately normal if the sample size is large enough, regardless of the population's distribution.

Question 2

A researcher collects a random sample of 40 home prices from a right-skewed population. Which condition allows the use of the Central Limit Theorem to analyze the sample mean?

Accepted Answer

The sample size is large (n \u003e 25), so the sampling distribution of the mean is approximately normal.

Question 3

Given a population with mean $$\mu = 100$$ and standard deviation $$\sigma = 20$$, what is the standard error of the mean for samples of size $$n = 25$$?

Accepted Answer

$$\frac{20}{\sqrt{25}} = 4$$

Question 4

A sample of 30 songs has a mean length of 250 seconds and a sample standard deviation of 85 seconds. What is the 95% confidence interval for the mean song length? (Use $$t^* = 2.042$$ for 29 degrees of freedom.)

Accepted Answer

$$250 \pm 2.042 	imes \frac{85}{\sqrt{30}}$$

Question 5

Which of the following is NOT a condition that must be checked before constructing a confidence interval for a population mean?

Accepted Answer

The population must be binomially distributed.

Question 6

A 95% confidence interval for the mean speed of cars on a highway is $$(61.35, 65.25). $$What does this interval mean?

Accepted Answer

We are 95% confident that the true mean speed of all cars on the highway is between 61.35 and 65.25 mph.

Question 7

When should you use a t-distribution instead of a z-distribution for inference about a mean?

Accepted Answer

When the population standard deviation $$\sigma is $$unknown.

Question 8

A hypothesis test for the mean years of experience among nurses yields a p-value of 0.019 with $$\alpha = 0.05. $$What is the correct conclusion?

Accepted Answer

Reject the null hypothesis; there is evidence that the mean years of experience has changed.

Question 9

Which of the following scenarios describes a matched pairs (dependent samples) design?

Accepted Answer

Measuring students' test scores before and after a reading intervention.

Question 10

Suppose you construct a 95% confidence interval for the difference in mean home prices between two neighborhoods and the interval is $$(-270.73, 1.53). $$What does this indicate?

Accepted Answer

There is no significant difference in mean home prices between the two neighborhoods.

Question 11

Which formula gives the standard error for the difference between two independent sample means?

Accepted Answer

$$\sqrt{\frac{s_1$^2}{n_1$} + \frac{s_2$^2}{n_2$}}$$

Question 12

A researcher wants to test if the mean number of credits taken by day and evening students is different. Which is the correct null hypothesis?

Accepted Answer

$$H_0$$: \mu_{day} = \mu_{evening}$$

Question 13

If a 95% confidence interval for the mean difference in pre- and post-test scores after an intervention is $$(–2.90, –0.19)$$, what can you conclude?

Accepted Answer

There is a significant improvement in scores after the intervention.

Question 14

Which of the following is true about the t-distribution as the sample size increases?

Accepted Answer

It approaches the normal distribution.

Question 15

Why is it generally recommended to use the unpooled version of the two-sample t-test?

Accepted Answer

It is more accurate in most situations than the pooled version.

Question 16

A sample of 35 students has a mean of 4.14 movies watched per month with a standard deviation of 10.02. What is the standard error of the mean?

Accepted Answer

$$\frac{10.02}{\sqrt{35}}$$

Question 17

Which of the following best explains why a 95% confidence interval is wider than a 90% confidence interval for the same data?

Accepted Answer

A higher confidence level requires a larger margin of error.

Question 18

If the p-value in a hypothesis test is greater than the significance level $$\alpha$$, what is the correct conclusion?

Accepted Answer

Fail to reject the null hypothesis.

Question 19

Which of the following is a correct interpretation of a two-sided hypothesis test and a confidence interval for the same data?

Accepted Answer

They will always lead to the same conclusion about the null hypothesis.

Chapter 9: Inferring Population Means – Confidence Intervals and Hypothesis Tests

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