Question

Grades on a standardized test are known to have a mean of 940 for students in the United States. The test is administered to 436 randomly selected students in​ Florida; in this​ sample, the mean is 952.22 and the standard deviation ​(s​) is 101.52.

The​ 95% confidence interval for the average test score for Florida students is ​(__ , __). ​(Round your responses to two decimal places.​)

Answer 1

Solution:

Given that,

   = 952.22 ....... Sample mean

s = 101.52 ........Sample standard deviation

n = 436 ....... Sample size

Note that, Population standard deviation() is unknown. So we use t distribution.

Our aim is to construct 95% confidence interval.

c = 0.95

= 1- c = 1- 0.95 = 0.05

  /2 = 0.05 2 = 0.025

Also, d.f = n - 1 = 436 - 1 = 435

    =    =  _0.025,435 = 1.965

( use t table or t calculator to find this value..)

The margin of error is given by

E =  /2,d.f. * ( / n )

=  1.965 * (101.52 / 436 )

= 9.55

Now , confidence interval for mean() is given by:

( - E ) <   <  ( + E)

(952.22 - 9.55)   <   <  (952.22 + 9.55)

942.67 <   < 961.77

Required interval is (942.67 , 961.77)

Answer :

The​ 95% confidence interval for the average test score for Florida students is (942.67 , 961.77)