Let’s solve this step by step based on the given image.

(a) Identify the claim and state $H_0$ and $H_a$

Key Observations:

• The problem states that the hospital claims the standard deviation has decreased.
• Variance is represented by $\sigma_1^2$ (initial process) and $\sigma_2^2$ (new process).

Null Hypothesis ($H_0$):

The variance has not decreased: $H_0: \sigma_1^2 \leq \sigma_2^2$

Alternative Hypothesis ($H_a$):

The variance has decreased: $H_a: \sigma_1^2 > \sigma_2^2$

From the options, the correct answer is B.

(b) Determine the critical value and rejection region.

Given:

• $\alpha = 0.10$ (significance level).
• Critical $F_0 = 1.67$ (from the problem).

Rejection Region:

Since this is a right-tailed test, the rejection region is: $F > F_0$

The correct answer is A: $F > F_0$.

(c) Calculate the test statistic $F$.

The formula for the $F$-statistic is: $F = \frac{s_1^2}{s_2^2}$

Given:

• Variance for the initial process: $s_1^2 = 0.36$.
• Variance for the new process: $s_2^2 = 0.16$.

Calculation:

$F = \frac{0.36}{0.16} = 2.25$

So, $F = 2.25$ (rounded to two decimal places).

Final Steps:

1. Compare the test statistic $F = 2.25$ with the critical value $F_0 = 1.67$.
2. Since $F > F_0$, reject the null hypothesis $H_0$.

This means there is sufficient evidence to support the hospital's claim that the standard deviation has decreased.

Let me know if you need additional clarification or help! 😊

Related Questions:

1. What is the relationship between the $F$-distribution and the variance ratio test?
2. How does changing $\alpha$ (significance level) affect the critical value?
3. What are the assumptions behind conducting an $F$-test?
4. Can $F$-tests be used for two-tailed tests, and if so, how?
5. How do sample sizes affect the $F$-statistic's critical value?

Tip:

Always verify that the variances are independent and the populations are normally distributed when using an $F$-test.