To solve this hypothesis test, we need to follow these steps:

Given Data:

• Sample size ($n$) = 23
• Sample mean ($\bar{x}$) = 57
• Sample standard deviation ($s$) = 3.44
• Null hypothesis ($H_0$): $\mu \leq 55.1$
• Alternative hypothesis ($H_1$): $\mu > 55.1$
• Significance level ($\alpha$) = 0.1

a) This is a right-tailed test.

• Correct: Since $H_1$ is $\mu > 55.1$, it is indeed a right-tailed test.

b) Calculate the test statistic:

The test statistic for the t-test is calculated using the formula: $t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}$ Where:

• $\bar{x}$ = sample mean = 57
• $\mu_0$ = population mean under the null hypothesis = 55.1
• $s$ = sample standard deviation = 3.44
• $n$ = sample size = 23

Substitute the values: $t = \frac{57 - 55.1}{3.44 / \sqrt{23}}$

First, calculate the standard error (SE): $SE = \frac{3.44}{\sqrt{23}} \approx \frac{3.44}{4.795} \approx 0.718$

Now, calculate the t-value: $t = \frac{57 - 55.1}{0.718} \approx \frac{1.9}{0.718} \approx 2.646$

Therefore, the correct test statistic rounded to three decimal places is: $t \approx 2.646$

c) At $\alpha = 0.1$, the rejection region:

• We need to find the critical value for $t$ with $\alpha = 0.1$ and degrees of freedom $df = n - 1 = 23 - 1 = 22$.

Using the t-distribution table or a calculator for $t_{0.1, 22}$, we find the critical value: $t_{0.1, 22} \approx 1.321$

So, the rejection region for this right-tailed test is: $t > 1.321$

d) The decision is to:

• Since our calculated $t$-value $2.646$ falls in the rejection region $t > 1.321$, we reject the null hypothesis $H_0$.

Therefore, the correct decision is: $\text{reject } H_0 \text{ since the test statistic falls in the rejection region.}$

Summary:

• a) Correct
• b) $t = 2.646$
• c) $t > 1.321$
• d) reject $H_0$

Would you like more details or have any questions about this hypothesis test?

Here are 8 related questions for further exploration:

1. What is the formula for calculating the t-statistic in a one-sample t-test?
2. How do you determine the degrees of freedom in a t-test?
3. What is the significance level ($\alpha$) in hypothesis testing, and how is it used?
4. How do you interpret the p-value in the context of hypothesis testing?
5. What are the assumptions underlying the use of a t-test?
6. How does sample size affect the outcome of a t-test?
7. What is the difference between a one-tailed and a two-tailed test?
8. How do you use a t-distribution table to find critical values?

Tip: Always double-check your calculations and ensure you are using the correct formula and critical values for your hypothesis test.