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 \(H_{a}: \mu_{1} \neq \mu_{2}\) using the paired difference sample results \(\bar{x}_{d}=-2.6, s_{d}=4.1\) \(n_{d}=18\)

Short answer

The short answer will be determined at Step 5, either 'Reject H0, there's a significant difference between the two means.' or 'Do not reject H0, there's no evidence of a significant difference between the two means.'

Step-by-step solution

Calculate the test statistic

The test statistic (T0) for a paired t-test is the sample mean difference divided by the standard error of the difference. The standard error (SE) is the sample standard deviation divided by the square root of the sample size. Given the notation, \( \bar{x}_{d} \), the sample mean difference;\( s_{d} \), the sample standard deviation of the differences;\( n_{d} \), the sample size; the test statistic can be calculated asT0 = \( \frac{\bar{x}_{d}}{s_{d} / \sqrt{n_{d}}}\)

Apply the provided values

By filling in the provided values into the formula from Step 1, we get T0 = \( \frac{-2.6}{4.1 / \sqrt{18}} \)

Compute the test statistic

Compute the value by following the operations order (division before multiplication). This will be the computed T0 value.

Determine the critical t-value

To test the null hypothesis at the \( \alpha=0.05 \) level, look up the critical t-value for a two-tailed test with \( n_{d}-1 = 18 - 1 = 17 \) degrees of freedom. As a standard, the critical t-value for a two-tailed test with \( \alpha=0.05 \) and 17 degrees of freedom from the t-distribution table is approximately ± 2.093.

Compare T0 and the critical t-value

Compare the computed T0 value with the critical t-value. If the absolute value of T0 is greater than the critical t-value, reject the null hypothesis. If it's less, do not reject the null hypothesis.

Interpret the result

The interpretation depends on the result from Step 5. If you rejected H0, you can say that there's a significant difference between the means. If you didn't reject H0, report that there's no evidence of a significant difference.