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	89	46	135
B	96	49	145
C	78	76	154
Column Total	263	171	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 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 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?

YesNo    

What sampling distribution will you use?

chi-squareuniform    normalStudent's tbinomial

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 < 0.010p-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 = 0.05

The null and alternate hypotheses.

H0: Ceremonial ranking and pottery type are independent.

H1: Ceremonial ranking and pottery type are not independent.

(b)

Ceremonial Ranking	Cooking Jar Sherds	Decorated Jar Sherds (Noncooking)	Row Total
A	89	46	135
B	96	49	145
C	78	76	154
Column Total	263	171	434
Expected Frq= (RT*CT)/GT	O= observed	E=Expected
RT=	Row Total
CT=	Column Total
GT=	434
E1=	135*263/434
E1=	81.80875576
	Calculation table for Chi-Square
	O	E	O-E	(O-E)2/E
	89	81.8087558	7.19124424	0.63213275
	46	53.1912442	-7.19124424	0.972227562
	96	87.8686636	8.13133641	0.75247112
	49	57.1313364	-8.13133641	1.157309384
	78	93.3225806	-15.3225806	2.515805671
	76	60.6774194	15.3225806	3.869338547
SUM=	434	434		9.899285034
Chi-Square =	sum of	(O-E)2/E
Chi-Square =	9.899

Test Statistics = 9.899285034

Are all the expected frequencies greater than 5?

Yes

What sampling distribution will you use?

Chi-Square

Degrees of freedom = (r-1)(c-1)

(3-1)(2-1)

= 2

The P-Value = 0.00708

0.005 < p-value < 0.010

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

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