Question

The following table shows site type and type of pottery for a random sample of 628 sherds at an archaeological location.

	Pottery Type
Site Type	Mesa Verde Black-on-White	McElmo Black-on-White	Mancos Black-on-White	Row Total
Mesa Top	77	62	50	189
Cliff-Talus	83	75	55	213
Canyon Bench	92	72	62	226
Column Total	252	209	167	628

Use a chi-square test to determine if site type and pottery type are independent at the 0.01 level of significance.

(a) What is the level of significance?


State the null and alternate hypotheses.

H₀: Site type and pottery are not independent.
H₁: Site type and pottery are independent.

H₀: Site type and pottery are independent.
H₁: Site type and pottery are independent.  

  H₀: Site type and pottery are not independent.
H₁: Site type and pottery are not independent.

H₀: Site type and pottery are independent.
H₁: Site type and pottery are not independent.

(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?

What sampling distribution will you use?

uniform

Student's t    

normal

binomial

chi-square

What are the degrees of freedom?


(c) Find or estimate the P-value of the sample test statistic. (Round your answer to three decimal places.)

p-value > 0.1000

.050 < p-value < 0.100   

0.025 < p-value < 0.0500

.010 < p-value < 0.0250

.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 1% level of significance, there is sufficient evidence to conclude that site and pottery type are not independent.At the 1% level of significance, there is insufficient evidence to conclude that site and pottery type are not independent.    

Answer 1

Solution:

Here, we have to use chi square test for independence of two categorical variables.

(a) What is the level of significance?

We are given level of significance = α = 0.01

State the null and alternate hypotheses.

Null hypothesis: H₀: Site type and pottery are independent.

Alternative hypothesis: H₁: Site type and pottery are not independent.

(b) Find the value of the chi-square statistic for the sample.

Test statistic formula is given as below:

Chi square = ∑[(O – E)^2/E]

Where, O is observed frequencies and E is expected frequencies.

E = row total * column total / Grand total

α = 0.01

Critical value = 13.2767

(by using Chi square table or excel)

Calculation tables for test statistic are given as below:

Observed Frequencies (O)
	Pottery Type
Site type	Messa	McElmo	Mancos	Total
Mesa top	77	62	50	189
Cliff-Talus	83	75	55	213
Canyon Bench	92	72	62	226
Total	252	209	167	628

Expected Frequencies (E)
	Pottery Type
Site type	Messa	McElmo	Mancos	Total
Mesa top	75.84076	62.89968	50.25955	189
Cliff-Talus	85.47134	70.88694	56.64172	213
Canyon Bench	90.6879	75.21338	60.09873	226
Total	252	209	167	628

Calculations
(O - E)
1.159236	-0.89968	-0.25955
-2.47134	4.113057	-1.64172
1.312102	-3.21338	1.901274

(O - E)^2/E
0.017719	0.012869	0.00134
0.071457	0.238651	0.047584
0.018984	0.137287	0.060148

Chi square = ∑[(O – E)^2/E] = 0.606039

Chi square statistic = 0.606

Are all the expected frequencies greater than 5?

Yes, all expected frequencies are greater than 5.

What sampling distribution will you use?

chi-square

What are the degrees of freedom?

We are given

Number of rows = r = 3

Number of columns = c = 3

Degrees of freedom = df = (r – 1)*(c – 1) = 2*2 = 4

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

P-value = 0.96239

(By using Chi square table or excel)

p-value > 0.1000

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

P-value > α = 0.01

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

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

At the 1% level of significance, there is insufficient evidence to conclude that site and pottery type are not independent.    

There is sufficient evidence to conclude that Site type and pottery are independent.