Question

Researchers measured skulls from different time periods in an attempt to determine whether interbreeding of cultures occurred. Results are given below. Assume that both samples are independent simple random samples from populations having normal distributions. Use a significance level to test the claim that the variation of maximal skull breadths in 4000 B.C. is the same as the variation in A.D. 150.

n x s
4000 B.C 30 131.64 mm 5.11 mm
A.D. 150 30 136.21 mm 5.32 mm

Required:
a. What are the null and alternative hypotheses?
b. Identify the test statistic?
c. Use technology to identify the P-value?
d. What is the conclusion for this hypothesis test?

Answer 1

Following are the solution to the given choices:

Explanation:

Given value:

`s_1 = 5.11\\s_2 = 5.32\\n_1 = 30\\n_2 = 30\\`

In point a:

Null hypothesis:

`H_0: \sigma_1^2=\sigma_2^2`

VS

Alternative:

`H_a:\sigma_1^2\\eq \sigma_2^2`

In point b:

Testing the statistic:

`F=(s_2^2)/(s_1^2) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ( s_2 > S_1)`

`=((5.32)^2)/((5.11)^2)\\\\=(28.3024)/(26.1121)\\\\=1.0838`

In point c:

Calculating the p-value, and the chances of rejection of
`H_o`:

`\to p-value=P( Reject\ H_o)`

`=P(F_(29,29)>1.0838)\\\\= 0.4194 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{using statistical table by excel}.`

The
`p-value > \alpha = 0.05`,
`H_o` is not rejected at the level of
`5\%`.

In point d:

Higher H0 must not be refused. It's indeed clear that perhaps the claim that the shift in maximum skull sizes is 4000 B.C is supported by sufficient evidence. The variation in A.D is the same. 150.