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%	924
Obsidian	10.6%	151
Welded Tuff	11.4%	166
Pedernal chert	13.1%	190
Other	3.6%	55

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

chi-square    

uniform

binomial

Student's t

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

The statistical software output for this problem is:

Chi-Square goodness-of-fit results:
Observed: Oi
Expected: Ei

N	DF	Chi-Square	P-value
1486	4	0.67994815	0.9538

Observed	Expected
924	910.918
151	157.516
166	169.404
190	194.666
55	53.496

Hence,

a) Level of significance = 0.01

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

b) Test statistic = 0.680

Yes

Chi - square

Degrees of freedom = 4

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.