3. Scores this year on the SAT mathematics test (SAT-M) for students taking the

Question

3. Scores this year on the SAT mathematics test (SAT-M) for students taking the test for the first time are believed to be Normally distributed with mean mu 1. For students taking the test for the second time, this year's score are also believed to be Normally distributed hut with a possibly different mean mu 2. We wish to estimate the difference mu2 - mu1. A random sample of the SAT-M scores of 100 students who took the test for the first time this year was obtained, and the mean of these 100 scores was x1 = 504.5, while the standard deviation was s1 = 100. A random sample of the SAT-M scores of 30 students who took the test for the second time this year was also obtained, and the mean of these 30 scores was x2 = 539.1 while the standard deviation was s2 = 90. A 95% confidence interval for mu2 - mu1 is (use the conservative value for the degrees of freedom) 34.6 ? 37.70. 34.6 ? 32.68. 34.6 ? 39.34.

Explanation / Answer

alpha,a = 0.05
n1 = 100 , xbar1 = 504.5 , s1 = 100
n2 = 30 , xbar2 = 539.1 , s2 = 90

standard error, SE = sqrt((s1^2/n1)+(s2^2/n2))
SE = sqrt((100^2/100) + (90^2/30)) = 19.23538

degree of freedom = min(n1-1, n2-1) = min(99,29)
= 29

t(a/2,df) = t(0.025, 29) = 2.045

confidence interval:
(xbar2-xbar1) +/- [t(a/2,n1+n2-2) * SE]
(539.1-504.5) +/- [2.045 * 19.23538]
34.6 +/- 39.34

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