Question

The state test scores for 12 randomly selected high school seniors are shown on the right. Complete parts​ (a) through​ (c) below.

Assume the population is normally distributed.

12 scores-

1430 1220 985 694 729 837 724 750 544 621 1443 949

1) find the sample mean

2) find the standard deviation

3) construct a 90% confidence interval for the population mean

Answer 1

Solution :

Given that,

x	x2
143	20449
1220	1488400
985	970225
694	481636
729	531441
837	700569
724	524176
750	562500
544	295936
621	385641
1443	2082249
949	900601
∑x=9639	∑x2=8943823

Mean ˉx=∑xn

=143+1220+985+694+729+837+724+750+544+621+1443+949/12

=9639/12

=803.25

Sample Standard deviation S=√∑x2-(∑x)2nn-1

=√8943823-(9639)212/11

=√8943823-7742526.75/11

=√1201296.25/11

=√109208.75

=330.4675

Degrees of freedom = df = n - 1 = 12 - 1 = 11

At 90% confidence level the z is ,

= 1 - 90% = 1 - 0.90 = 0.10

/ 2 = 0.10 / 2 = 0.05

t _/2,df = t_0.05,11=1.796

Margin of error = E = t_/2,df * (s /n)

= 1.796 * (330.47 / 12 )

=171.32

Margin of error = 171.32

The 90% confidence interval estimate of the population mean is,

- E <  < + E

803.25 - 171.32 < < 803.25 + 171.32

631.92 < < 974.57

(631.92, 974.57 )