Question

A transect is an archaeological study area that is 1/5 mile wide and 1 mile long. A site in a transect is the location of a significant archaeological find. Let x represent the number of sites per transect. In a section of Chaco Canyon, a large number of transects showed that x has a population variance σ2 = 42.3. In a different section of Chaco Canyon, a random sample of 28 transects gave a sample variance s2 = 48.3 for the number of sites per transect.

Required:
Use a 5% level of significance to test the claim that the variance in the new section is greater than 42.3.

Answer 1

To test the claim that the variance in the new section is greater than 42.3, a chi-square test can be used. The test statistic is calculated and compared to the critical value, resulting in the rejection of the null hypothesis and the conclusion that the variance in the new section is greater than 42.3.

Step-by-step explanation:

To test the claim that the variance in the new section is greater than 42.3, we can use a chi-square test. We will set up the null and alternative hypotheses as follows:

Null hypothesis (H0): The variance in the new section is equal to 42.3 (σ2 = 42.3)

Alternative hypothesis (Ha): The variance in the new section is greater than 42.3 (σ2 > 42.3)

We will use a 5% significance level, which corresponds to a chi-square critical value of approximately 39.3 with 27 degrees of freedom (df = n - 1).

To calculate the test statistic, we will use the sample variance (s2) obtained from the random sample of 28 transects. The formula for the chi-square test statistic for variance is:

χ2 = (n - 1) * s2 / σ2

Substituting the values, we get:

χ2 = (28 - 1) * 48.3 / 42.3 ≈ 31.75

Comparing this test statistic to the critical value, we can see that χ2 > 39.3, which means that the test statistic falls in the rejection region. Therefore, we reject the null hypothesis and conclude that the variance in the new section is indeed greater than 42.3.

Answer 2

There is significant evidence to conclude that carina e in New section is greater than 42.3

Step-by-step explanation:

Given :

Sample variance, s² = 48.3

Population variance, σ² = 42.3

Sample size, n = 28

α = 0.05

The hypothesis :

H0 : σ² = 42.3

H0 : σ² > 42.3

The test statistic, χ² : (n-1)*s²/σ²

χ² = [(28 - 1) * 48.3] / 42.3

χ² = (27 * 48.3) / 42.3

χ² = 1304.1 / 42.3

χ² = 30.829787

χ² = 30.830

The Critical value at α = 0.05 ; df = (28-1) = 27

Critical value(0.05, 27) = 27.587

Reject H0 if χ² statistic > Critical value

Since, 30.830 > 27.587 ; Reject H0 ; and conclude that variance in the new section is greater than 42.3