Answer :
Answer:
90% confidence interval for the population mean test score is [95.40 , 106.59]
Step-by-step explanation:
We are given that the population mean score for mathematics test is 115. It is known that the population standard deviation of test scores is 17.
Also, a random sample of 25 students take the exam. The mean score for this group is 101.
The, pivotal quantity for 90% confidence interval for the population mean test score is given by;
P.Q. = [tex]\frac{Xbar-\mu}{\frac{\sigma}{\sqrt{n} } }[/tex] ~ N(0,1)
where, X bar = sample mean = 101
[tex]\sigma[/tex] = population standard deviation
n = sample size = 25
So, 90% confidence interval for the population mean test score, [tex]\mu[/tex] is ;
P(-1.6449 < N(0,1) < 1.6449) = 0.90
P(-1.6449 < [tex]\frac{Xbar-\mu}{\frac{\sigma}{\sqrt{n} } }[/tex] < 1.6449) = 0.90
P(-1.6449 * [tex]\frac{\sigma}{\sqrt{n} }[/tex] < [tex]{Xbar-\mu}[/tex] < 1.6449 * [tex]\frac{\sigma}{\sqrt{n} }[/tex] ) = 0.90
P(X bar - 1.6449 * [tex]\frac{\sigma}{\sqrt{n} }[/tex] < [tex]\mu[/tex] < X bar + 1.6449 * [tex]\frac{\sigma}{\sqrt{n} }[/tex] ) = 0.90
90% confidence interval for [tex]\mu[/tex] = [ X bar - 1.6449 * [tex]\frac{\sigma}{\sqrt{n} }[/tex] , X bar + 1.6449 * [tex]\frac{\sigma}{\sqrt{n} }[/tex] ]
= [ 101 - 1.6449 * [tex]\frac{17}{\sqrt{25} }[/tex] , 10 + 1.6449 * [tex]\frac{17}{\sqrt{25} }[/tex] ]
= [95.40 , 106.59]
Therefore, 90% confidence interval for the population mean test score is [95.40 , 106.59] .