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	86	49	135
B	87	58	145
C	74	80	154
Column Total	247	187	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 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 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?

normalbinomial     uniformchi-squareStudent'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 > 0.1000.050 < 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)

Level of Significance = = 0.05
Correct option:

H0: Ceremonial ranking and pottery are independent.

H1: Ceremonial ranking and pottery are dependent.

(b)

(i)

Expected Frequencies are got as follows:

Ceremonial ranking	Cooking for Sherds	Decorated Jar Sherds	Row Total
A	255X135/434=79.3203	55.6797	135
B	85.1959	59.8041	145
C	90.4839	63.5161	154
Column total	255	179	434

Test Statistic is got as follows:

Observed (O)	Expected (E)	(O - E)2/E
81	79.3203	0.0356
54	55.6797	0.0507
96	85.1959	1.3701
49	59.8041	1.9518
78	90.4839	1.7224
76	63.5161	2.4537
Total = =		7.5843

Test Statistic is: 7.584

(ii)

Correct option:

Yes

(iii)

Correct option:

chi square

(iv)

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

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

                            = 2

(c) By Technology, P - Value = 0.0225

So,

Correct option:

0.010 < p - value < 0.025

(d)

Since P - Value is less than , Reject null hypothesis.

So,

Correct option:

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

(e)

Correct option:

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