Question

Two college instructors are interested in whether or not there is any variation in the way they grade math exams. They each grade the same set of 12 exams. The first instructor's grades have a variance of 52.3. The second instructor's grades have a variance of 89.9. Test the claim that the first instructor's variance is smaller. The level of significance is 5%.

Answer 1

Given that,
population variance (σ^2) =52.3
sample size (n) = 12
sample variance (s^2)=89.9
null, Ho: σ^2 =52.3
alternate, H1 : σ^2 <52.3
level of significance, α = 0.05
from standard normal table,left tailed ᴪ^2 α/2 =19.675
since our test is left-tailed
reject Ho, if ᴪ^2 o < -19.675
we use test statistic chisquare ᴪ^2 =(n-1)*s^2/o^2
ᴪ^2 cal=(12 - 1 ) * 89.9 / 52.3 = 11*89.9/52.3 = 18.908
| ᴪ^2 cal | =18.908
critical value
the value of |ᴪ^2 α| at los 0.05 with d.f (n-1)=11 is 19.675
we got | ᴪ^2| =18.908 & | ᴪ^2 α | =19.675
make decision
hence value of | ᴪ^2 cal | < | ᴪ^2 α | and here we do not reject Ho
ᴪ^2 p_value =0.0628
ANSWERS
---------------
null, Ho: σ^2 =52.3
alternate, H1 : σ^2 <52.3
test statistic: 18.908
critical value: -19.675
p-value:0.0628
decision: do not reject Ho
we do not have enough evidence to support the claim that the first instructor's variance is smaller.