Step 1: Calculate the test statistic (z)

The test statistic for a one-sample $z$-test is calculated using the formula:

$z = \frac{M - \mu}{\sigma / \sqrt{n}}$

where:

• $M = 93.4$ (sample mean),
• $\mu = 80.2$ (hypothesized population mean),
• $\sigma = 19.2$ (population standard deviation),
• $n = 19$ (sample size).

Substituting these values:

$z = \frac{93.4 - 80.2}{19.2 / \sqrt{19}}$

Calculating the denominator first:

$\sigma / \sqrt{n} = \frac{19.2}{\sqrt{19}} \approx 4.404$

Then, calculating the entire expression:

$z \approx \frac{13.2}{4.404} \approx 2.998$

So, the test statistic is approximately:

$\text{test statistic} = 2.998$

Step 2: Find the p-value

Since this is a one-tailed test (with $H_a: \mu > 80.2$), we find the probability that $Z \geq 2.998$.

Using standard normal distribution tables or a calculator, we find:

$\text{p-value} \approx 0.0014$

Step 3: Compare the p-value to $\alpha$

Given:

• $\alpha = 0.002$,
• $\text{p-value} = 0.0014$.

Since the p-value (0.0014) is less than $\alpha$ (0.002), we conclude:

$\text{The p-value is less than (or equal to) } \alpha$

Step 4: Make a decision

Since the p-value is less than $\alpha$, we reject the null hypothesis.

Final Conclusion

Since we reject the null hypothesis, we conclude:

$\text{The sample data support the claim that the population mean is greater than 80.2.}$

Summary of Results

• Test Statistic: 2.998
• p-value: 0.0014
• Decision: Reject the null hypothesis
• Conclusion: The sample data support the claim that the population mean is greater than 80.2.

Would you like further details on any part of the solution?

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Related Questions:

1. How does increasing the sample size affect the test statistic and p-value in hypothesis testing?
2. What would the conclusion be if the significance level were 0.001 instead of 0.002?
3. How would the test statistic and p-value change if the sample mean were lower, say 85?
4. Why is the assumption of a normally distributed population important in this context?
5. How does using a smaller significance level impact the likelihood of rejecting the null hypothesis?

Tip:

Always double-check your significance level ($\alpha$) before making a decision; small changes in $\alpha$ can alter the outcome in hypothesis testing.