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 \(95 \%\) confidence interval for \(\mu_{1}-\mu_{2}\) using the paired data in the following table: $$ \begin{array}{lcc} \hline \text { Case } & \text { Situation 1 } & \text { Situation 2 } \\ \hline 1 & 77 & 85 \\ 2 & 81 & 84 \\ 3 & 94 & 91 \\ 4 & 62 & 78 \\ 5 & 70 & 77 \\ 6 & 71 & 61 \\ 7 & 85 & 88 \\ 8 & 90 & 91 \\ \hline \end{array} $$

Short answer

Based on the provided data and assuming the conditions are met, the best estimate for \(\mu_{1}-\mu_{2}\) is the computed mean difference (\(\overline{d}\)), the computed value as per step 4 is the margin of error, and the lower and upper bounds found in step 5 form the 95% confidence interval.

Step-by-step solution

Compute the differences

As the data is paired, we first need to compute the difference \(d = x_{1} - x_{2}\) for each row or case in the given table. By doing this we are effectively creating a new set of numbers.

Compute the summary statistics

Once the differences are calculated, find the mean difference (\(d\)) and the standard deviation of the differences, as these will be used to compute the confidence interval.

Find the t-score from table

To compute the confidence interval, we need a t-score, which can be found in any standard statistical reference under the t-distribution table with two-sided values. Since we have 7 degrees of freedom (8 pairs minus 1) and want a 95% confidence interval, the relevant t-score from the t-distribution table is approximately 2.365.

Compute the margin of error

The margin of error is computed using the formula \(\overline{d} \pm \) (t-score * standard error). Where standard error is the standard deviation divided by the square root of the sample size, in this case 8.

Observe the confidence interval

Finally, we use the computed margin of error to observe the confidence interval for the difference in means. Subtract and add the margin of error from your observed \(\overline{d}\) (mean difference) to find the lower and upper bounds of the 95% confidence interval, respectively.