Question

Use a t-distribution and the given matched pair sample results to complete the test of the given hypotheses. Assume the results come from random samples, and if the sample sizes are small, assume the underlying distribution of the differences is relatively normal. Assume that differences are computed using \(d=x_{1}-x_{2}\). Test \(H_{0}: \mu_{1}=\mu_{2}\) vs \(H_{a}: \mu_{1}>\mu_{2}\) using the paired data in the following table: $$ \begin{array}{lllllllllll} \hline \text { Situation } & 1 & 125 & 156 & 132 & 175 & 153 & 148 & 180 & 135 & 168 & 157 \\ \text { Situation } & 2 & 120 & 145 & 142 & 150 & 160 & 148 & 160 & 142 & 162 & 150 \\ \hline \end{array} $$

Short answer

The conclusion on whether to accept or reject the null hypothesis is based on the comparison of the computed t-score to the critical t-value. The exact numerical result is dependent on calculations which are not included here.

Step-by-step solution

Calculate the differences

Start by computing the differences for each matched pair using the formula \(d = x_{1} - x_{2}\). The computed differences will then be used for all subsequent calculations.

Calculate the sample mean and sample standard deviation

Next, calculate the sample mean of the computed differences to derive the average difference. The mean difference can be computed using the formula \(\mu_{d} = (1/n) \sum d \), where \(n\) represents the number of pairs. Compute the sample standard deviation of the computed differences using the formula \(\sigma_{d} = \sqrt{(1/n-1) \sum (d - \mu_{d})^{2}}\), to measure the average variability in the differences.

Calculate the t-score

Compute the t-score using the formula, \(t = \mu_{d} / (\sigma_{d} / \sqrt{n})\). This value represents how extreme the sample mean is and will later be compared to the critical t-value.

Determine the critical t-value

To accept or reject the null hypothesis, find the critical t-value that corresponds to the level of significance (typically 0.05) from a t-distribution table. This value is based on the degrees of freedom, which in this case is given by the formula \(df = n-1\). The critical t-value is the threshold t-value beyond which we reject the null hypothesis.

Compare the t-score with the critical t-value

Finally, compare the computed t-score with the critical t-value. If the computed t-value is greater than the critical t-value, the null hypothesis (\(H_{0}:\mu_{1}=\mu_{2}\)) is rejected in favor of the alternative hypothesis (\(H_{a}:\mu_{1}>\mu_{2}\)). If the t-score is less than or equal to the critical t-value, the null hypothesis is not rejected.