You wish to test the following claim (HaHa) at a significance level of α=0.02α=0.02.

      Ho:μ=86.2Ho:μ=86.2
      Ha:μ<86.2Ha:μ<86.2

You believe the population is normally distributed, but you do not know the standard deviation. You obtain a sample of size n=31n=31 with a mean of M=80.3M=80.3 and a standard deviation of SD=11.4SD=11.4.

What is the critical value for this test? (Report answer accurate to three decimal places.)
critical value = Incorrect

What is the test statistic for this sample? (Report answer accurate to three decimal places.)
test statistic = Correct

The test statistic is...

in the critical region

not in the critical region

Correct


This test statistic leads to a decision to...

reject the null

accept the null

fail to reject the null

Correct


As such, the final conclusion is that...

There is sufficient evidence to warrant rejection of the claim that the population mean is less than 86.2.

There is not sufficient evidence to warrant rejection of the claim that the population mean is less than 86.2.

The sample data support the claim that the population mean is less than 86.2.

There is not sufficient sample evidence to support the claim that the population mean is less than 86.2.

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.

Learn how to conduct a one-sample t-test for hypothesis testing with a detailed example. This problem involves testing a claim about the population mean at a significance level of α=0.02, with a sample mean of M=80.3 and standard deviation of SD=11.4. Find the critical value, calculate the test statistic, and interpret the results.

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