Question

The following table shows ceremonial ranking and type of pottery sherd for a random sample of 434 sherds at an archaeological location.

Ceremonial Ranking	Cooking Jar Sherds	Decorated Jar Sherds (Noncooking)	Row Total
A	90	45	135
B	96	49	145
C	81	73	154
Column Total	267	167	434

Use a chi-square test to determine if ceremonial ranking and pottery type are independent at the 0.05 level of significance.

(a) What is the level of significance?


State the null and alternate hypotheses.

H₀: Ceremonial ranking and pottery type are independent.
H₁: Ceremonial ranking and pottery type are not independent.H₀: Ceremonial ranking and pottery type are independent.
H₁: Ceremonial ranking and pottery type are independent.    H₀: Ceremonial ranking and pottery type are not independent.
H₁: Ceremonial ranking and pottery type are not independent.H₀: Ceremonial ranking and pottery type are not independent.
H₁: Ceremonial ranking and pottery type are 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?

Yes

No    

What sampling distribution will you use?

Student's t

chi-square    

binomial

uniform

normal

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 > 0.100

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 5% level of significance, there is sufficient evidence to conclude that ceremonial ranking and pottery type are not independent.

At the 5% level of significance, there is insufficient evidence to conclude that ceremonial ranking and pottery type are not independent.    

Answer 1

Answer a)

Level of significance = 0.05

The null and alternate hypotheses are :

H1: Ceremonial ranking and pottery type are not independent.

H0: Ceremonial ranking and pottery type are independent.

Answer b)

Expected Frequencies are calculated as follows:

Ceremonial Ranking	Cooking Jar Sherds	Decorated Jar Sherds (Noncooking)
A	=(135*267)/434 = 83.0530	=(135*167)/434 = 51.9470
B	=(145*267)/434 = 89.2051	=(145*167)/434 = 55.7949
C	=(154*267)/434 = 94.7419	=(154*167)/434 = 59.2581

Contribution table is as follows:

Ceremonial Ranking	Cooking Jar Sherds	Decorated Jar Sherds (Noncooking)
A	=(90-83.0530)^2/83.0530 = 0.5811	=(45-51.9470)^2/51.9470 = 0.9290
B	=(96-89.2051)^2/89.2051 = 0.5176	=(49-55.7949)^2/55.7949 = 0.8275
C	=(81-94.7419)^2/94.7419 = 1.9932	=(73-59.2581)^2/59.2581 = 3.1868

Chi-square statistic = Sum of all contributions = 0.5811+0.5176+1.9932+0.9290+0.8275+3.1868

Chi-square statistic = 8.035

Yes, all the expected frequencies are greater than 5. (Refer to expected values table)

The sampling distribution used will be chi-square.

The degrees of freedom = (r-1)*(c-1) = (3-1)*(2-1) = 2

P-value corresponding to Chi-square statistic = 8.035 and df = 2 is obtained using p-value calculator. Screenshot below:

P-value = 0.0180

So, correct answer is 0.010 < p-value < 0.025

Answer d)

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

Answer e)

At the 5% level of significance, there is sufficient evidence to conclude that ceremonial ranking and pottery type are not independent.