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	97	48	145
C	83	71	154
Column Total	270	164	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 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 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.

(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?

chi-square

binomial    normal

uniform

Student's t

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

(a) The level of significance is given for the Chi-square test of independence between categorical variable ceremonial ranking and pottery type.

Null and alternative hypothesis:

Ceremonial ranking and pottery type are independent.

Ceremonial ranking and pottery type are not independent.

(b) The chi-square statistic is given by-    

So, first we need to find the Expected frequency(E) for the given Observed data.

Expected frequency table:

Ceremonial Rank	Cooking Jar Sherds	Decorated Jar Sherds (Noncooking)
A
B
C

The Chi-square statistic is calculated :

• Yes all the expected frequencies are greater than 5, we can verify it by looking at the expected frequency table.
• The sampling distribution used is : Chi-square
• Degrees of freedom,df=(row-1)(column-1)=(3-1)(2-1)=2*1=2, i.e., df=2

(c) P-value:

The p-value is calculated as upto 3 decimal places is:

(d) Decision:

Since,

(e) So, at 5% level of significance , there is sufficient evidence to support the alternative hypothesis, H₁ .

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