Do students tend to improve their SAT Mathematics (SAT-M)scores the second time they take the test? A random sample of four students who took the test twice received the following scores. Assume that the change in SAT-M score (second score - first score)for the population of all students taking the test twice is Normally distributed.What is a 90% confidence interval for $\mu$,the mean change in SAT-M score?
A)(-8.24,58.24)
B)(-18.08,68.08)
C)(-22.56,72.56)
D)(-39.31,89.31)

Question

Do students tend to improve their SAT Mathematics (SAT-M)scores the second time they take the test? A random sample of four students who took the test twice received the following scores.  Assume that the change in SAT-M score (second score - first score)for the population of all students taking the test twice is Normally distributed.What is a 90% confidence interval for $\mu$,the mean change in SAT-M score?
A)(-8.24,58.24) 
B)(-18.08,68.08) 
C)(-22.56,72.56) 
D)(-39.31,89.31)

Quizplus · Accepted Answer

We need to use the formula for a confidence interval for a population mean:

CI = sample mean +/- (t-value)(standard error)

First, we need to find the sample mean and standard error:

sample mean = (11+43+6+12)/4 = 18

standard deviation = sqrt((11-18)^2 + (43-18)^2 + (6-18)^2 + (12-18)^2)/3 = 16.81

standard error = standard deviation/sqrt(n) = 16.81/sqrt(4) = 8.405

Next, we need to find the t-value. Since we have a sample size of 4, we have 3 degrees of freedom. Using a t-table or calculator, we find that the t-value for a 90% confidence interval with 3 degrees of freedom is 2.353.

Plugging in all the values, we get:

CI = 18 +/- (2.353)(8.405) = (-22.56, 72.56)

Therefore, the best choice is (C), as it is the only choice that contains the correct confidence interval.

Do Students Tend to Improve Their SAT Mathematics (SAT-M)scores the Second