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 19 transects gave a sample variance s² = 51.5 for the number of sites per transect. Use a 5% level of significance to test the claim that the variance in the new section is greater than 42.3.

(a) What is the level of significance?


State the null and alternate hypotheses.

(a) H₀: σ² = 42.3; H₁: σ² < 42.3

(b) H₀: σ² = 42.3; H₁: σ² > 42.3   

(c) H₀: σ² > 42.3; H₁: σ² = 42.3

(d) H₀: σ² = 42.3; H₁: σ² ≠ 42.3

(b) Find the value of the chi-square statistic for the sample. (Round your answer to two decimal places.)


What are the degrees of freedom?


What assumptions are you making about the original distribution?

(a) We assume a uniform population distribution.

(b) We assume a exponential population distribution.    

(c) We assume a normal population distribution.

(d) We assume a binomial population distribution.

(c) Find or estimate the P-value of the sample test statistic.

(a) P-value > 0.1000.

(b) 050 < P-value < 0.100  

(c) 0.025 < P-value < 0.050

(d) 0.010 < P-value < 0.025

(e) 0.005 < P-value < 0.010P-value < 0.005

(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis?

(a) Since the P-value > α, we fail to reject the null hypothesis.

(b) Since the P-value > α, we reject the null hypothesis.    

(c) Since the P-value ≤ α, we reject the null hypothesis.

(d) Since the P-value ≤ α, we fail to reject the null hypothesis.

(e) Interpret your conclusion in the context of the application.

(a) At the 5% level of significance, there is insufficient evidence to conclude that the variance is greater in the new section.

(b) At the 5% level of significance, there is sufficient evidence to conclude that the variance is greater in the new section.    

Answer 1

a)
0.05

Below are the null and alternative Hypothesis,
Null Hypothesis, H0: σ^2 = 42.3
Alternative Hypothesis, Ha: σ^2 > 42.3

b)
Test statistic,
ꭓ^2 = (n-1)*s^2/σ^2
ꭓ^2 = (19 - 1)*51.5/42.3
ꭓ^2 = 21.91

df = 19 - 1 = 18

We assume a normal population distribution.

c)
P-value Approach
P-value = 0.236

option A

d)
) Since the P-value > α, we fail to reject the null hypothesis.

e)
At the 5% level of significance, there is insufficient evidence to conclude that the variance is greater in the new section