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	89	56	145
C	76	78	154
Column Total	255	179	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 not 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 not independent.   

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

H₀: Ceremonial ranking and pottery type are 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?

YesNo    

What sampling distribution will you use?

Student's t

chi-square

   binomial

normal

uniform

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 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

Solution:

Given:

Observed Frequencies
	type of pottery sherd
Ceremonial Ranking	Cooking Jar Sherds	Decorated Jar Sherds (Noncooking)	Row Total
A	90	45	135
B	89	56	145
C	76	78	154
Column Total	255	179	434

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

Part (a) What is the level of significance?

level of significance = 0.05

State the null and alternate hypotheses.

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

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

Formula for Chi square independence

Where

O_ij = Observed frequencies for i^th row and j^th column.

E_ij = Expected frequencies for i^th row and j^th column.

Where

Thus Expected frequency table is:

Expected Frequencies
	type of pottery sherd
Ceremonial Ranking	Cooking Jar Sherds	Decorated Jar Sherds (Noncooking)
A	79.320	55.680
B	85.196	59.804
C	90.484	63.516

Thus we get:

Oij	Eij	Oij^2/Eij
90	79.320	102.118
45	55.680	36.369
89	85.196	92.974
56	59.804	52.438
76	90.484	63.835
78	63.516	95.787

Thus

Are all the expected frequencies greater than 5?

Yes.

What sampling distribution will you use?

chi-square

What are the degrees of freedom?

df = ( R - 1) X (C - 1)

R = Number of Rows = 3

C = Number of Columns = 2

thus

df = ( 3- 1 )X ( 2-1)

df = 2 X 1

df = 2

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

Look in Chi-square table for df = 2 row and find the interval in which fall. then find corresponding right tail area

fall between 9.210 and 10.597

thus p-value is between 0.005 and 0.010

Thus correct answer is:

0.005 < p-value < 0.010

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

Since 0.005 < p-value < 0.010 that means p-value < 0.05 level of significance.

Thus correct answer is:

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

Part 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.