Answer:
(a) 95% confidence interval for the mean score of all New Jersey third graders is (53.01, 56.99)
(b) 90% confidence interval for the difference in mean scores between lowa and New Jersey is (3.71, 4.29)
Explanation:
Confidence Interval = mean + or - Error margin (E)
(a) New Jersey
mean = 55
sd = 10
n = 99
degree of freedom = n - 1 = 99 - 1 = 98
Confidence level (C) = 95% = 0.95
Significance level = 1 - C = 1 - 0.95 = 0.05 = 5%
t-value corresponding to 98 degrees of freedom and 5% significance level is 1.9846
E = t×sd/√n = 1.9846×10/√99 = 1.99
Lower limit = mean - E = 55 - 1.99 = 53.01
Upper limit = mean + E = 55 + 1.99 = (56.99)
95% confidence interval is between 53.01 and 56.99.
(b) Lowa
mean = 59
sd = 13
n = 198
Difference in mean between Lowa and New Jersey = 59 - 55 = 4
Difference in sd between Lowa and New Jersey = 13 - 10 = 3
degree of freedom = n1 + n2 - 2 = 99 + 198 - 2 = 295
Confidence level = 90%
Significance level = 10%
t-value corresponding to 295 degrees of freedom and 10% confidence interval is 1.65312
E = t×sd/√n = 1.65312×3/√297 = 0.29
Lower limit = mean - E = 4 - 0.29 = 3.71
Upper limit = mean + E = 4 + 0.29 = 4.29
90% confidence interval is between 3.71 and 4.29
