Question

The types of raw materials used to construct stone tools found at an archaeological site are shown below. A random sample of 1486 stone tools were obtained from a current excavation site.

Raw Material	Regional Percent of Stone Tools	Observed Number of Tools as Current excavation Site
Basalt	61.3%	905
Obsidian	10.6%	163
Welded Tuff	11.4%	165
Pedernal chert	13.1%	201
Other	3.6%	52

Use a 1% level of significance to test the claim that the regional distribution of raw materials fits the distribution at the current excavation site.

(a) What is the level of significance?


State the null and alternate hypotheses.

H₀: The distributions are the same.
H₁: The distributions are the same.

H₀: The distributions are the same.
H₁: The distributions are different.    

H₀: The distributions are different.
H₁: The distributions are different.

H₀: The distributions are different.
H₁: The distributions are the same.

(b) Find the value of the chi-square statistic for the sample. (Round the expected frequencies to at least three decimal places. Round the test statistic to three decimal places.)


Are all the expected frequencies greater than 5?

Yes

No    

What sampling distribution will you use?

normal

Student's t    

binomial

uniform

chi-square

What are the degrees of freedom?


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

P-value > 0.100

0.050 < P-value < 0.100    

0.025 < P-value < 0.050

0.010 < P-value < 0.025

0.005 < P-value < 0.010

P-value < 0.005

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

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

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

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

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

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

At the 0.01 level of significance, the evidence is sufficient to conclude that the regional distribution of raw materials does not fit the distribution at the current excavation site.

At the 0.01 level of significance, the evidence is insufficient to conclude that the regional distribution of raw materials does not fit the distribution at the current excavation site.

Answer 1

Solution:-

a)

H₀: The distributions are the same.
H₁: The distributions are different.

b)

Formulate an analysis plan. For this analysis, the significance level is 0.01. Using sample data, we will conduct a chi-square goodness of fit test of the null hypothesis.

Analyze sample data. Applying the chi-square goodness of fit test to sample data, we compute the degrees of freedom, the expected frequency counts, and the chi-square test statistic. Based on the chi-square statistic and the degrees of freedom, we determine the P-value.

DF = k - 1 = 5 - 1
D.F = 4
(E_i) = n * p_i


X² = 0.592

where DF is the degrees of freedom, k is the number of levels of the categorical variable, n is the number of observations in the sample, E_i is the expected frequency count for level i, O_i is the observed frequency count for level i, and X² is the chi-square test statistic.

The P-value is the probability that a chi-square statistic having 1 degrees of freedom is more extreme than 0.592.

We use the Chi-Square Distribution Calculator to find P(X² > 0.592) = 0.964.

Interpret results. Since the P-value (0.964) is greater than the significance level (0.05), we cannot accept the null hypothesis.

c) P-value > 0.100.

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

e) At the 0.01 level of significance, the evidence is insufficient to conclude that the regional distribution of raw materials does not fit the distribution at the current excavation site.