Let's break down the problem step by step to find the correct critical value, test statistic, and final conclusion.

1. Identify the Hypotheses

• Null Hypothesis ($H_0$): $\mu = 86.2$
• Alternative Hypothesis ($H_a$): $\mu < 86.2$ (left-tailed test)

2. Given Information

• Significance level: $\alpha = 0.02$
• Sample size: $n = 31$
• Sample mean: $M = 80.3$
• Sample standard deviation: $SD = 11.4$
• Population is normally distributed but the standard deviation is unknown.

3. Determine the Critical Value

Since the population standard deviation is unknown and the sample size is small ($n < 30$), we use the t-distribution to find the critical value.

• Degrees of Freedom: $df = n - 1 = 31 - 1 = 30$
• Significance Level: $\alpha = 0.02$
• This is a left-tailed test, so we are looking for the critical value $t_\alpha$ where the cumulative probability is $0.02$ in the left tail.

Using a t-distribution table or calculator:

$t_{0.02, 30} \approx -2.042$

So, the critical value is approximately $-2.042$.

4. Calculate the Test Statistic

The test statistic is calculated using the following formula:

$t = \frac{M - \mu}{\frac{SD}{\sqrt{n}}}$

Substituting the given values:

$t = \frac{80.3 - 86.2}{\frac{11.4}{\sqrt{31}}} = \frac{-5.9}{\frac{11.4}{5.57}} = \frac{-5.9}{2.046} \approx -2.882$

The test statistic is approximately $-2.882$.

5. Compare Test Statistic to Critical Value

• Critical value: $-2.042$
• Test statistic: $-2.882$

Since $-2.882$ is less than $-2.042$, the test statistic falls in the critical region.

6. Decision and Conclusion

• Since the test statistic is in the critical region, we reject the null hypothesis.
• Conclusion: There is sufficient evidence to support the claim that the population mean is less than 86.2.

Summary:

• Critical Value: $-2.042$
• Test Statistic: $-2.882$
• Decision: Reject the null hypothesis.
• Final Conclusion: There is sufficient evidence to support the claim that the population mean is less than 86.2.

Do you want details or have any questions?

Here are some related questions:

1. What happens if the significance level is changed?
2. How would the critical value change with a different sample size?
3. What if the test was two-tailed instead of one-tailed?
4. How does the test statistic compare to a z-statistic in this context?
5. What is the impact of using a t-distribution instead of a z-distribution?
6. What role does the sample standard deviation play in the calculation?
7. How does sample size affect the degrees of freedom and critical value?
8. Can we reject the null hypothesis at a different significance level?

Tip: Always verify that the test is correctly identified as one-tailed or two-tailed, as this directly affects the critical value and the decision.