To determine if the population mean is greater than 100 at the $\alpha = 0.10$ level of significance, we will follow these steps:

Part 1: State the Hypotheses

The null hypothesis (H₀) and the alternative hypothesis (H₁) are:

• Null Hypothesis (H₀): $\mu = 100$
• Alternative Hypothesis (H₁): $\mu > 100$

This is a one-tailed test.

Part 2: Compute the Test Statistic

We use the following formula for the test statistic $t$:

$t = \frac{\bar{x} - \mu}{s / \sqrt{n}}$

Where:

• $\bar{x} = 107.5$ (sample mean)
• $\mu = 100$ (population mean under the null hypothesis)
• $s = 17.5$ (sample standard deviation)
• $n = 40$ (sample size)

Let's calculate the test statistic $t$:

First, calculate the standard error of the mean (SEM):

$\text{SEM} = \frac{17.5}{\sqrt{40}} = \frac{17.5}{6.3246} \approx 2.767$

Now, calculate the test statistic $t$:

$t = \frac{107.5 - 100}{2.767} \approx \frac{7.5}{2.767} \approx 2.71$

Thus, the test statistic $t_0$ is approximately 2.71.

Summary

• Hypotheses: $H_0: \mu = 100$, $H_1: \mu > 100$
• Test Statistic: $t_0 \approx 2.71$

Would you like more details or have any questions?

Here are five questions you can ask next:

1. How do we determine the critical value for a one-tailed t-test?
2. What is the p-value associated with our test statistic?
3. How do we interpret the results of this t-test?
4. What assumptions must be met to use a t-test in this scenario?
5. How can we calculate the confidence interval for the population mean?

Tip: Always verify that your sample size is sufficiently large to justify using the t-distribution, especially when dealing with small sample sizes.