You wish to test the following claim (
H
a
) at a significance level of
α
=
0.01
.

H
o
:
μ
=
60.7

H
a
:
μ
≠
60.7

You believe the population is normally distributed, but you do not know the standard deviation. You obtain a sample of size
n
=
293
with a mean of
M
=
59.6
and a standard deviation of
S
D
=
5.3
.

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

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

The test statistic is...
in the critical region
not in the critical region

This test statistic leads to a decision to...
reject the null
accept the null
fail to reject the null

As such, the final conclusion is that...
There is sufficient evidence to warrant rejection of the claim that the population mean is not equal to 60.7.
There is not sufficient evidence to warrant rejection of the claim that the population mean is not equal to 60.7.
The sample data support the claim that the population mean is not equal to 60.7.
There is not sufficient sample evidence to support the claim that the population mean is not equal to 60.7.

To solve this hypothesis testing problem, we need to follow several steps. Let's go through them one by one.

### 1. Determine the critical value
Since the population standard deviation is unknown and the sample size \(n = 293\) is large, we will use the **t-distribution** to find the critical value.

Given:
- Significance level \( \alpha = 0.01 \)
- The test is two-tailed.

For a two-tailed test at a significance level of \( \alpha = 0.01 \), we need to find the critical value \( t_{\alpha/2} \) using the degrees of freedom \( df = n - 1 = 293 - 1 = 292 \).

Using a t-distribution table or statistical software, the critical value for \( df = 292 \) at \( \alpha/2 = 0.005 \) is approximately:

\[
\text{critical value} = \pm 2.580
\]

### 2. Calculate the test statistic
The test statistic for a one-sample t-test is calculated using the formula:

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

Where:
- \( M = 59.6 \) (sample mean)
- \( \mu = 60.7 \) (population mean under null hypothesis)
- \( SD = 5.3 \) (sample standard deviation)
- \( n = 293 \) (sample size)

Plugging in the values:

\[
t = \frac{59.6 - 60.7}{\frac{5.3}{\sqrt{293}}}
\]
\[
t = \frac{-1.1}{\frac{5.3}{17.1175}}
\]
\[
t = \frac{-1.1}{0.3097} \approx -3.552
\]

### 3. Compare the test statistic with the critical value
Now, we compare the calculated test statistic with the critical value:

- The test statistic \( t = -3.552 \)
- The critical value \( \pm 2.580 \)

Since \( -3.552 \) is less than \( -2.580 \), the test statistic falls **in the critical region**.

### 4. Decision
Because the test statistic is in the critical region, we **reject the null hypothesis**.

### 5. Conclusion
Since we reject the null hypothesis, the conclusion is:

"There is sufficient evidence to warrant rejection of the claim that the population mean is equal to 60.7."

### Summary:
- **Critical value**: \( \pm 2.580 \)
- **Test statistic**: \( -3.552 \)
- **Test statistic is in the critical region**: Yes
- **Decision**: Reject the null hypothesis
- **Conclusion**: There is sufficient evidence to warrant rejection of the claim that the population mean is equal to 60.7.

Would you like more details, or do you have any questions about this? Here are 5 related questions that might help deepen your understanding:

1. What assumptions are necessary for the t-test to be valid in this context?
2. How would the conclusion change if the sample size were much smaller?
3. What is the difference between a one-tailed and two-tailed test?
4. How would you interpret the p-value associated with this test statistic?
5. What would be the impact on the test result if the significance level was changed to 0.05?

**Tip**: Always verify if your test statistic calculation matches the context of the hypothesis test (one-tailed vs. two-tailed).

Learn how to conduct a hypothesis test for a population mean using a one-sample t-test. Understand how to determine the critical value and calculate the test statistic with a real-world example. Suitable for college-level students and researchers.

Math Problem Statement

You wish to test the following claim ( H a ) at a significance level of α

H o : μ

You believe the population is normally distributed, but you do not know the standard deviation. You obtain a sample of size n

293 with a mean of M

59.6 and a standard deviation of S D

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