To construct a 99% confidence interval for the paired data sets $A$ and $B$, we will follow these steps:

1. Calculate the differences between corresponding values of $A$ and $B$.
2. Find the mean of these differences.
3. Calculate the standard deviation of these differences.
4. Use the t-distribution to find the critical value for a 99% confidence level.
5. Construct the confidence interval.

Let's solve it step by step.

Step 1: Calculate the differences

The data sets are:

• $A$: 5.8, 6.8, 8.7, 5.7, 5.8
• $B$: 8.2, 7.1, 7.0, 6.9, 8.3

Calculate the differences $D = A - B$: $D = 5.8 - 8.2, \, 6.8 - 7.1, \, 8.7 - 7.0, \, 5.7 - 6.9, \, 5.8 - 8.3$ $D = -2.4, -0.3, 1.7, -1.2, -2.5$

Step 2: Mean of the differences

$\bar{D} = \frac{-2.4 + (-0.3) + 1.7 + (-1.2) + (-2.5)}{5} = \frac{-4.7}{5} = -0.94$

Step 3: Standard deviation of the differences

Calculate the standard deviation $s_D$: $s_D = \sqrt{\frac{1}{n-1} \sum (D_i - \bar{D})^2}$ Where $n = 5$.

Step 4: Critical value for a 99% confidence level

Use the t-distribution with $n-1 = 4$ degrees of freedom.

Step 5: Construct the confidence interval

The confidence interval is given by: $\bar{D} \pm t^* \left(\frac{s_D}{\sqrt{n}}\right)$ Where $t^*$ is the critical value.

Let me calculate these values for you.The calculations yield the following results:

• Mean of the differences: $\bar{D} = -0.94$
• Standard deviation of the differences: $s_D = 1.733$
• Critical value for a 99% confidence level: $t^* = 4.604$

The 99% confidence interval for the mean difference is: $(-4.508, 2.628)$

Therefore, the correct answer is option B: $(-4.508, 2.628)$.

Would you like to delve into any of the details or have more questions?

Related Questions:

1. What is the formula for calculating the standard deviation of paired differences?
2. How does the confidence level affect the width of a confidence interval?
3. Why is the t-distribution used instead of the z-distribution in this case?
4. How would the confidence interval change if we had a larger sample size?
5. What are some practical applications of paired sample confidence intervals?

Tip:

When working with small sample sizes (typically $n < 30$), it's better to use the t-distribution rather than the z-distribution to account for the additional uncertainty in the sample estimate of the standard deviation.