(a) A 95% confidence interval for the mean score of all New Jersey third graders is [56.41, 59.59]
.
(b) A 90% confidence interval for the difference in mean scores between Iowa and New Jersey is [3.363, 4.637] .
(c) Yes, we are 90% confident that the population means for Iowa and New Jersey students are different.
Step-by-step explanation:
We are given that a new standardized test is given to 100 randomly selected third-grade students in New Jersey. The sample average score Y on the test is 58 points and the sample standard deviation sY is 8 points.
(a) Firstly, the pivotal quantity for finding the confidence interval for the population mean is given by;
P.Q. = ~
where, = sample average score = 58 points
s = sample standard deviation = 8 points
n = sample of third-grade students = 100
= population mean score of all New Jersey third graders
Here for constructing a 95% confidence interval we have used a One-sample t-test statistics because we don't know about population standard deviation.
So, a 95% confidence interval for the population mean, is;
P(-1.987 < < 1.987) = 0.95 {As the critical value of t at 99 degrees of
freedom are -1.987 & 1.987 with P = 2.5%} P(-1.987 < < 1.987) = 0.95
P( < < ) = 0.95
P( < < ) = 0.95
95% confidence interval for = [ , ]
= [ , ]
= [56.41, 59.59]
Therefore, a 95% confidence interval for the mean score of all New Jersey third graders is [56.41, 59.59]
.
Now, the same test is given to 200 randomly selected third graders from Iowa, producing a sample average of 62 points and a sample standard deviation of 11 points.
(b) Firstly, the pivotal quantity for finding the confidence interval for the difference in population mean is given by;
P.Q. = ~
where,
= = 3.163
Here for constructing a 90% confidence interval we have used a two-sample t-test statistics because we don't know about population standard deviations.
So, a 90% confidence interval for the difference in two population means, () is;
P(-1.645 < < 1.645) = 0.90 {As the critical value of t at 298 degrees of
freedom are -1.645 & 1.645 with P = 5%} P(-1.645 < < 1.645) = 0.90
90% confidence interval for () = [ , ]
= [ , ]
= [3.363, 4.637]
Therefore, a 90% confidence interval for the difference in mean scores between Iowa and New Jersey is [3.363, 4.637] .
(c) Yes, we are 90% confident that the population means for Iowa and New Jersey students are different because in the above interval 0 is not included.