Do SAT scores for high school students differ depending on the students' intended eld of study? Fifteen students who intended to major in engineering were compared with 15 students who intended to major in language and literature. Given in the accompanying table are the means and standard deviations of the scores on the mathematics portion of the SAT for the two groups of students: Standard Deviation Lanauaaanitaraaaa The objective is to construct a condence interval for m M to answer the research question, where m (respectively, #2) is the mean (mathematics) SAT score of all engineering (respectively, language/literature) majors. Let Tm denote the random variable with m degrees of freedom. Note that 13(ng 5 2.04841) = 0.975 and P(T14 5 2.14479) = 0.975, (a) Following the method outlined in Slide 21 of Chapter 9, construct a 95% condence interval for #1 #2. Slide 21 Chapter 9: Case 3: Pooled T Condence Interval Then the 100(1 a)% condence interval for A is A A 1 1 A i tgnn-In-Z UP = X _ Y i t,m+n2 SP ; + E The 100(1 a)% upper condence bound for A is A A 1 1 A + ta,m+n20p = X Y + ta,m+n2 Sp ; + a The 100(1 00% lower condence bound for A is A A 1 1 A _ ta,m+n20p = X Y _ ta,m+n2 Sp 5 + a \f(c) Among the three confidence intervals you constructed, which one has the shortest width and why?2 (m- 1)S , O 2 ' - 1 X - y Silk/m + In