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	91	44	135
B	87	58	145
C	81	73	154
Column Total	259	175	434

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

State the null and alternate hypotheses.

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 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 or 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. (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

The table for observed frequency is ,

Ceremonial Ranking	Cookig Jar Shards	Decorated Jar Sherds	Total
A	91	44	135
B	87	58	145
C	81	73	154
Total	259	175	434

The table for expected frequency is ,

Ceremonial Ranking	Cookig Jar Shards	Decorated Jar Sherds
A	80.56451613	54.43548387
B	86.53225806	58.46774194
C	91.90322581	62.09677419

The table for is ,

Ceremonial Ranking	Cookig Jar Shards	Decorated Jar Sherds
A	102.7871872	35.56503704
B	87.47027027	57.536
C	71.39031239	85.81766234

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

b) The value of the chi-square test statistic is ,

Here , All expected frequencies are greater than 5

Here , to use the Chi-square sampling distribution

The degrees of freedom are ,

df=(r-1)*(c-1)=(3-1)*(2-1)=2

c) P-value = ; From excel "=CHIDIST(E10,2)"

Here , 0.025 < p-value < 0.050

d) Decision : Here , P-value= 0.038 < 0.05

Therefore , reject Ho

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