Question

A box contains billions of tickets, all labeled either 1, 2, 3, 4, or 5. A simple random sample of 400 tickets is taken; the sample mean is 2.6 and the sample SD is 1. Let μ represent the population mean. Which of the following statements are correct? (More than one statement may be correct.)

A. There is a 68% probability that μ is between 2.55 and 2.65.
B. The interval between 2.55 and 2.65 is a 68% confidence interval for μ.

C. There is a 90% probability that μ is between 2.5 and 2.7.
D. The interval between 2.5 and 2.7 is a 90% confidence interval for μ.

E. The interval between 2.5 and 2.7 is a 95% confidence interval for μ.

F. There is a 95% probability that μ is between 2.5 and 2.7.

Answer 1

A. is correct

B. is correct

Note :

The 68% confidence interval for is

where

degrees of freedom = n-1= 400-1 =399

For 68% confidence , with df =399, tc = 1.00 (from t table)

Thus , 68% confidence interval for is

= (2.55, 2.65 )

Thus we can say that , P( 2.55 < < 2.65) =68% : A is correct

Also , 68% confidence interval for is (2.55, 2.65 ) : B is correct

C. is not correct

D.is not correct

Note : For 90% confidence , with df =399, tc = 1.65 (from t table)

Thus , 90% confidence interval for is

= ( 2.52, 2.68 )

E. is correct

F.is correct

Note : For 95% confidence , with df =399, tc = 1.97 (from t table)

Thus , 90% confidence interval for is

= ( 2.5, 2.7 )

Thus we can say that , P( 2.5 < < 2.7) =95% : F is correct

Also , 95% confidence interval for is (2.5, 2.7 ) : E  is correct