Let's break this down step by step.

(a) Hypotheses

You've already correctly identified the hypotheses as:

✅ H0: 𝜇d = 0 versus Ha: 𝜇d > 0

This is a right-tailed paired difference t-test to detect if the population mean difference is greater than zero.

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(b) Conduct the Test

Given:

• Sample size: $n = 17$
• Mean of the differences: $d = 5.3$
• Variance of the differences: $s_d^2 = 257$
• Significance level: $\alpha = 0.05$

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Step 1: Calculate the Standard Deviation

The standard deviation is the square root of the variance:

$s_d = \sqrt{257} = 16.0312$

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Step 2: Test Statistic Formula

The t-statistic for a paired t-test is:

$t = \frac{d}{s_d / \sqrt{n}}$

Substitute the values:

$t = \frac{5.3}{16.0312 / \sqrt{17}}$

$t = \frac{5.3}{16.0312 / 4.1231}$

$t = \frac{5.3}{3.888}$

$t \approx 1.363$

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Step 3: Conclusion

Now, compare the calculated t-value with the critical t-value at $\alpha = 0.05$ with $df = n - 1 = 16$.

Using a t-distribution table:

$t_{critical} = 1.746$

Since:

$t = 1.363 < t_{critical} = 1.746$

We fail to reject the null hypothesis.

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Final Conclusion:

There is not enough evidence to conclude that the population mean difference is greater than 0 at the 0.05 significance level.

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Do you want me to calculate the p-value or explain how to use technology (like Excel or TI-84) to solve this?

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Related Questions:

1. How do you find the p-value for this test?
2. What is the confidence interval for this paired t-test?
3. How does increasing the sample size affect the result?
4. What if the test was two-tailed instead of one-tailed?
5. How do you check the normality assumption in paired t-tests?

🔥 Tip:

Always check the normality assumption before performing paired t-tests, especially for small samples!