To construct the confidence interval for a hypothesis test involving the mean difference ($\mu_d$), we follow these steps:

1. Determine the Components:

• Sample Mean Difference ($\bar{d}$): Compute the mean of the differences between paired data.
• Standard Deviation of Differences ($s_d$): Compute the standard deviation of the paired differences.
• Sample Size ($n$): Count the number of pairs.
• Critical Value ($t^*$): Use the $t$-distribution critical value based on the confidence level and degrees of freedom ($df = n-1$).

2. Compute the Margin of Error (ME):

$\text{ME} = t^* \times \frac{s_d}{\sqrt{n}}$

3. Construct the Confidence Interval:

The confidence interval is: $\bar{d} - \text{ME} < \mu_d < \bar{d} + \text{ME}$

4. Relation to Hypothesis Test:

• If the confidence interval does not contain zero, it supports rejecting the null hypothesis ($H_0: \mu_d = 0$).
• If the confidence interval contains zero, it supports failing to reject the null hypothesis.

Example:

Assume the following values (to illustrate the calculation):

• $\bar{d} = 2.3$ years
• $s_d = 1.2$ years
• $n = 15$
• Confidence level: 95% ($t^*$ for $df = 14$ is approximately 2.145)

Calculate the Margin of Error:

$\text{ME} = 2.145 \times \frac{1.2}{\sqrt{15}} \approx 0.663$

Construct the Interval:

$2.3 - 0.7 < \mu_d < 2.3 + 0.7$ $1.6 < \mu_d < 3.0$

Final Confidence Interval:

1.6 years < $\mu_d$ < 3.0 years

This confidence interval does not include zero, so the same conclusion is reached as in part (a): reject the null hypothesis.

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Here are 5 related questions:

1. How do you compute the critical $t^*$ value for a confidence interval?
2. Why does the inclusion of zero in a confidence interval affect hypothesis test conclusions?
3. What is the significance of sample size in constructing confidence intervals?
4. How do we adjust the confidence interval for different confidence levels (e.g., 99%)?
5. How does variability in data affect the width of the confidence interval?

Tip: Always verify the degrees of freedom when using the $t$-distribution to ensure correct critical values.