QuestionEssays A professor wishes to see if two groups of students' essays differ in lengths, that is, the number of words in each essay. The professor randomly selects 12 essays from a group of students who are science majors and 10 essays from a group of humanities majors to compare. The data are shown. At $\alpha=0.05$, can it be concluded that there is a difference in the lengths of the essays between the two groups? Science majors \begin{tabular}{lllllllllllll} 2262 & 3162 & 2450 & 3533 & 2725 & 1776 & 2830 & 3765 & 3357 & 2356 & 3887 & 3416 \end{tabular}
Humanities majors $$\begin{array}{llllllllll} \hline 2604 & 2069 & 2123 & 1468 & 1952 & 2573 & 1886 & 2921 & 2237 & 2757 \end{array}$$
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Use $\mu_{1}$ for the mean of science majors and $\mu_{2}$ for the mean of humanities majors. Assume the populations are normally distributed, and that the variances are unequal.
Part 1 of 5 (a) State the hypotheses and identify the claim with the correct hypothesis. $$\begin{array}{l} H_{0}: \mu_{1}=\mu_{2} \text { not claim } \\ H_{1}: \mu_{1} \neq \mu_{2} \text { claim } \end{array}$$
This hypothesis test is a two-tailed $\quad$ test.
Part 2 of 5 (b) Find the critical value. Round the answer(s) to at least three decimal places. If there is more than one critical value, separate them with commas.
Critical value(s): $2.262,-2.262$
Part: $2 / 5$
Part 3 of 5 (c) Compute the test value. Round your answer to at least three decimal places. $$t=\square$$