Question

Use a t-distribution to find a confidence interval for the difference in means \(\mu_{1}-\mu_{2}\) using the relevant sample results from paired data. Give the best estimate for \(\mu_{1}-\) \(\mu_{2},\) the margin of error, and the confidence interval. Assume the results come from random samples from populations that are approximately normally distributed, and that differences are computed using \(d=x_{1}-x_{2}\) A \(99 \%\) confidence interval for \(\mu_{1}-\mu_{2}\) using the paired data in the following table:. $$ \begin{array}{lccccc} \hline \text { Case } & \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} \\ \hline \text { Treatment 1 } & 22 & 28 & 31 & 25 & 28 \\ \text { Treatment 2 } & 18 & 30 & 25 & 21 & 21 \\ \hline \end{array} $$

Short answer

The 99% confidence interval for \(\mu_{1}-\mu_{2}\) is given by \(\bar{d} \pm t \cdot \frac{s_{d}}{\sqrt{n}}\), where \(\bar{d}\) is the sample mean of the differences, \(t\) is the critical value from the t-distribution for a 99% confidence level, \(s_{d}\) is the sample standard deviation of the differences, and \(n\) is the number of pairs. The actual values depend on the computed mean, standard deviation, and the t-value from the t-distribution.

Step-by-step solution

Calculate Differences

Begin by calculating the difference for each pair in the data. That would mean subtracting each value of Treatment 2 from Treatment 1 to get the 'd' values.

Find Sample Mean and Standard Deviation

Once all the differences are calculated, compute the sample mean \(\bar{d}\) and the sample standard deviation \(s_{d}\). This will involve the formulas: \[\bar{d} = \frac{\sum{}d}{n}\] for the mean, where \(n\) is the number of pairs and \(\sum{}d\) is the sum of all 'd' values; and \[s_{d} = \sqrt{\frac{\sum{}(d - \bar{d})^2}{n-1}}\] for the standard deviation.

Identify Degrees of Freedom and t-value

The next step is to identify the degrees of freedom and the critical value from the t-distribution. The degrees of freedom are given by \(df = n - 1\). The t-value corresponds to the given 99% confidence level and can be retrieved from a t-distribution table or using statistical software.

Calculate Confidence Interval

Finally, using the t-value, calculate the confidence interval for the true mean difference using the formula: \[CI = \bar{d} \pm t \cdot \frac{s_{d}}{\sqrt{n}}\] where the term \(t \cdot \frac{s_{d}}{\sqrt{n}}\) refers to the margin of error.