Question

Given the following random sample from a normally distributed population, test the claim that the population has a standard deviation greater than 9.2, using a 5% level of significance. Show your work in the following steps: A) State the null and alternative hypotheses. B) Find the critical value(s). C) Calculate the test statistic. D) Make the decision to reject the null hypothesis or not. E) State the conclusion.

Sample:

51 53 65

47 63 73

35 54 64

57 64 44

66 53 61

47 37 59

42 50 56

52 50 42

42 67 67

69 45 77

Answer 1

Given that,
population standard deviation (σ)=9.2
sample standard deviation (s) =10.739
sample size (n) = 30
we calculate,
population variance (σ^2) =84.64
sample variance (s^2)=115.326121
null, Ho: σ =9.2
alternate, H1 : σ >9.2
level of significance, α = 0.05
from standard normal table,right tailed ᴪ^2 α/2 =42.557
since our test is right-tailed
we use test statistic chisquare ᴪ^2 =(n-1)*s^2/o^2
ᴪ^2 cal=(30 - 1 ) * 115.326121 / 84.64 = 29*115.326121/84.64 = 39.514
| ᴪ^2 cal | =39.514
critical value
the value of |ᴪ^2 α| at los 0.05 with d.f (n-1)=29 is 42.557
we got | ᴪ^2| =39.514 & | ᴪ^2 α | =42.557
make decision
hence value of | ᴪ^2 cal | < | ᴪ^2 α | and here we do not reject Ho
ᴪ^2 p_value =0.0922
ANSWERS
---------------
A.
null, Ho: σ =9.2
alternate, H1 : σ >9.2
C.
test statistic: 39.514
B.
critical value: 42.557
p-value:0.0922
D.
decision: do not reject Ho
E.
we do not have enough evidence to support the claim that the population has a standard deviation greater than 9.2.