Step 1 :State the Null and Alternative Hypotheses. The null hypothesis is that the means are equal, and the alternative hypothesis is that the means are not equal. This corresponds to option B. \( \mu_{1} \neq \mu_{2} \)
Step 2 :Compute the test statistic. Given values are \( n_{1} = 68, x_{1} = 2.76, s_{1} = 2.21, n_{2} = 72, x_{2} = 2.88, s_{2} = 2.43 \). The test statistic is calculated as \( t = \frac{x_{1} - x_{2}}{\sqrt{\frac{s_{1}^{2}}{n_{1}} + \frac{s_{2}^{2}}{n_{2}}}} \). The computed test statistic is approximately -0.3060
Step 3 :Compute the P-value. The degrees of freedom is calculated as \( df = n_{1} + n_{2} - 2 \). The p-value is then calculated using the test statistic and the degrees of freedom. The computed p-value is approximately 0.76
Step 4 :\(\boxed{\text{Final Answer:}}\) 1) The correct option is B. \( \mu_{1} \neq \mu_{2} \) 2) The test statistic is approximately -0.3060 3) The p-value is approximately 0.76