Week 8 Homework Score: \( 0.95 / 44 \quad 1 / 14 \) answered Question 2 Next Hypothesis Test for Difference in Population Means ( \( \sigma \) Unknown) You wish to test the following claim \( \left(H_{a}\right) \) at a significance level of \( \alpha=0 \) \[ \begin{array}{l} H_{o}: \mu_{1}=\mu_{2} \\ H_{a}: \mu_{1}<\mu_{2} \end{array} \] You believe both populations are normally distributed, but you do not kno either. We will assume that the population variances are not equal. You obtain a sample of size \( n_{1}=12 \) with a mean of \( M_{1}=88.6 \) and a sta from the first population. You obtain a sample of size \( n_{2}=10 \) with a mea deviation of \( S D_{2}=8.7 \) from the second population. What is the test statistic for this sample? (Report answer accurate to three test statistic \( = \) What is the \( \mathrm{p} \)-value for this sample? For this calculation, use the conservat degrees of freedom. The degrees of freedom is the minimum of \( n_{1}-1 \) and to four decimal places.) n-valio \( = \) - 1 Type here to search esc !? \( [* \) * * \&
Solution
Step 1 ：Step 1: Calculate the standard error of the difference in means: \( SE = \sqrt{\frac{S_{1}^{2}}{n_{1}} + \frac{S_{2}^{2}}{n_{2}}} \) where \( S_{1} = 14 \), \( n_{1} = 12 \), \( S_{2} = 8.7 \), and \( n_{2} = 10 \).
Step 2 ：Step 2: Calculate the test statistic using the formula: \( t = \frac{(\bar{x}_{1} - \bar{x}_{2}) - (\mu_{1} - \mu_{2})}{SE} \) where \( \bar{x}_{1} = 88.6 \), \( \bar{x}_{2} = 0 \), \( \mu_{1} - \mu_{2} = 0 \), and SE from Step 1.
Step 3 ：Step 3: Since we do not know the true population variances, use the conservative degrees of freedom: \( df = min(n_{1} - 1, n_{2} - 1) \) where \( n_{1} = 12 \) and \( n_{2} = 10 \).
Step 4 ：Step 4: Determine the corresponding p-value using the calculated t-statistic and degrees of freedom calculated in steps 2 and 3, respectively. To do this, find the corresponding area under the t-distribution curve to the right of the calculated t-statistic.
