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	88	47	135
B	94	51	145
C	74	80	154
Column Total	256	178	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 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.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?

chi-squarenormal    Student's tuniformbinomial

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

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

b)

Expected frequencies E_ij =

	Cooking Jar Sherds	CatDecorated Jar Sherds	Row Totals
A	88  (79.631)	47  (55.369)	135
B	94  (85.530)	51  (59.470)	145
C	74  (90.839)	80  (63.161)	154
Column Totals	256	178	434  (Grand Total)

The values in the brackets represent (expected values).

where O_ij = Observed value of two nominal variables
E_ij = Expected value of two nominal variables

by substituting all the values we get te chi-square statistic = 11.800

Yes all the expected frequencies greater than 5.

sampling distribution we use is chi-square

the degrees of freedom = (r-1)(c-1) = (3-1)*(2-1) = 2

c) p-value < 0.005

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

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