Math  /  Data & Statistics

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 α=0.05\alpha=0.05, can it be concluded that there is a difference in the lengths of the essays between the two groups?
Science majors 226231622450353327251776283037653357235638873416\begin{array}{lllllllllllll}2262 & 3162 & 2450 & 3533 & 2725 & 1776 & 2830 & 3765 & 3357 & 2356 & 3887 & 3416\end{array} Humanities majors \begin{tabular}{llllllllll} 2604 & 2069 & 2123 & 1468 & 1952 & 2573 & 1886 & 2921 & 2237 & 2757 \end{tabular}
Send data to Excel Use μ1\mu_{1} for the mean of science majors and μ2\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. H0:μ1=μ2 not claim H1:μ1μ2 claim \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: 1/51 / 5
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): \square

Studdy Solution
Determine the critical value(s) for the two-tailed test using the t-distribution table with the calculated degrees of freedom and α=0.05 \alpha = 0.05 .
For a two-tailed test, find the t-value such that the cumulative probability is α2=0.025 \frac{\alpha}{2} = 0.025 in each tail.
The critical value(s) are:
±tdf,0.025 \pm t_{df, 0.025}

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