Coaching companies claim that their courses can raise the SAT scores of high sch
ID: 3273695 • Letter: C
Question
Coaching companies claim that their courses can raise the SAT scores of high school students. But students who retake the SAT without paying for coaching also usually raise their scores A random sample of students who took the SAT twice found 427 who were coached and 2733 who were uncoached. Starting with their verbal scores on the first and second tries, we have these summary statistics: Use Table C to estimate a 99% confidence Interval for the mean gain of all students who are coached. 21.645 to 36.355 at 99% confidence. Now test the hypothesis that the score gain for coached students is greater than the score gain for uncoached students Let mu_1 be the score gain for all coached students. Let mu_2 be the score gain for uncoached students. (a) Give the alternative hypothesis: mu_1 - mu_2 > 0. (b) Give the t test statistic: 2.6459 (c) Give the appropriate critical value for alpha =5%: The conclusion is A. There is not sufficient evidence to support the claim that with coaching, the mean increase in scores is greater than without coaching. B. There is sufficient evidence to support the claim that with coaching, the mean increase In scores is greater than without coaching.Explanation / Answer
First we will calcualte 99% confidence interval for the mean gain of all students.
Here the values are given in table
99% confidence interval for the mean gain off all students who are coached
= Average gain for coached +- Z99% se [where seis the standard error of the gain coached here which is equal to = s/sqrt(n)]
= 29 +- 2.575 * (59/ sqrt(427))
= 29 +- 7.352
= (21.648, 36.352)
(a) Give the alternative hypotheis is 1 -2 > 0
(B) Test statistic
t = (xcoached -xuncoached)/ se0
where se0is the standard error of the gain difference here which is equal to = s/sqrt(n)
se0 =sp * sqrt [1/ncoached + 1/nuncoached]
where sp = sqrt [(ncoached -1)s2coach + (nnoncoached -1)s2noncoach )/ (ncoached + nuncoached -2)]
sp = sqrt [(426 * 592 + 2732 * 522 )/ 3158] = 53
se0 = 53 * sqrt [1/427 + 1/2733] = 2.758
t = (29 - 21)/ 2.758 = 2.9
(c) Appropriate critical value at alpha = 0.05 is 2.577
(d) There is sufficient claim to support the claim that with coaching, the mean increase in scores is greater than without coaching.
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