Q: Suppose a student times the travel of a bus for 25 days and finds a sample mean of 37 minutes and a sample standard deviation of 2.5 minutes. What is the 90% confidence interval for the true mean travel time? (Use $$t_{24} = 1.711)$$

$$(37 \pm 1.711 \times \frac{2.5}{\sqrt{25}})$$

Q: What does a 99% confidence interval of (5.88, 9.42) for a population mean $$c$$ mean?

We are 99% confident that the true population mean $$c is $$between 5.88 and 9.42.

Question 1

Which formula is used to calculate the standard error of the sample mean when the population standard deviation is unknown?

Accepted Answer

$$SE(\bar{y}) = \frac{s}{\sqrt{n}}$$

Question 2

A researcher collects a random sample of $$n = 20$$ and computes a 95% confidence interval for the population mean. Which model should be used to find the critical value?

Accepted Answer

t-model with $$df = 19$$

Question 3

Which of the following is NOT an assumption required for constructing a confidence interval for the population mean?

Accepted Answer

The population mean is known.

Question 4

Suppose a student times the travel of a bus for 25 days and finds a sample mean of 37 minutes and a sample standard deviation of 2.5 minutes. What is the 90% confidence interval for the true mean travel time? (Use $$t_{24} = 1.711)$$

Accepted Answer

$$(37 \pm 1.711 	imes \frac{2.5}{\sqrt{25}})$$

Question 5

What does a 99% confidence interval of (5.88, 9.42) for a population mean $$c$$ mean?

Accepted Answer

We are 99% confident that the true population mean $$c is $$between 5.88 and 9.42.

Question 6

Which property distinguishes the t-model from the standard Normal model when sample size is small?

Accepted Answer

The t-model has thicker tails.

Question 7

A hypothesis test for the population mean $$c is $$performed at significance level $$\alpha = 0.05. $$The p-value is 0.03. What is the correct decision?

Accepted Answer

Reject $$H_0$$ because the p-value is less than $$\alpha.$$

Question 8

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

Accepted Answer

$$t = \frac{\bar{y} - \mu_0$}{s / \sqrt{n}}$$

Question 9

If the underlying distribution of $$y is $$non-Normal or unknown, which condition allows us to use the t-model for inference?

Accepted Answer

The sample size is large.

Question 10

A coffee shop manager wants to test if the mean wait time is greater than 1.4 minutes. Which is the correct form of the alternative hypothesis?

Accepted Answer

$$H_A: \mu \u003e 1.4$$

Question 11

Why do we use the t-model instead of the standard Normal model when the population standard deviation is unknown?

Accepted Answer

Because the sample standard deviation $$s$$ varies from sample to sample, affecting the standard error.

Question 12

Which of the following best describes the degrees of freedom for a one-sample t-test?

Accepted Answer

$$df = n - 1$$

Question 13

If a 95% confidence interval for a mean is (12.2, 15.1), and you want to test $$H_0$$: \mu = 16$$ vs H_A$$: \mu 
eq 16$$ at $$\alpha = 0.05$$, what is the correct conclusion?

Accepted Answer

Reject $$H_0$$ because 16 is outside the interval.

Question 14

Which of the following statements about confidence intervals is correct?

Accepted Answer

A 95% confidence interval means that 95% of intervals constructed from repeated samples will contain the true mean.

Question 15

If the sample size increases, which of the following will decrease for a confidence interval for the mean?

Accepted Answer

The standard error

Question 16

In a one-sample t-test, what does the p-value represent?

Accepted Answer

The probability of observing a test statistic as extreme or more extreme, assuming the null hypothesis is true.

Question 17

Which of the following is a correct interpretation of a 90% confidence interval for the mean wait time at a coffee shop?

Accepted Answer

We are 90% confident that the true mean wait time is within the interval.

Question 18

If the degrees of freedom required for a t-table are not listed, what should you do?

Accepted Answer

Use the closest degrees of freedom.

Chapter 18: Inferences About Means – Confidence Intervals and Hypothesis Testing

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