Question

The following table shows site type and type of pottery for a random sample of 628 sherds at an archaeological location.

	Pottery Type
Site Type	Mesa Verde Black-on-White	McElmo Black-on-White	Mancos Black-on-White	Row Total
Mesa Top	76	63	50	189
Cliff-Talus	85	73	55	213
Canyon Bench	97	72	57	226
Column Total	258	208	162	628

Use a chi-square test to determine if site type and pottery type are independent at the 0.01 level of significance.

(a) What is the level of significance?


State the null and alternate hypotheses.

H₀: Site type and pottery are not independent.
H₁: Site type and pottery are not independent.

H₀: Site type and pottery are not independent.
H₁: Site type and pottery are independent.    

H₀: Site type and pottery are independent.
H₁: Site type and pottery are not independent.

H₀: Site type and pottery are independent.
H₁: Site type and pottery 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

No    

What sampling distribution will you use?

normal

Student's t    

uniform

binomial

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 1% level of significance, there is sufficient evidence to conclude that site and pottery type are not independent.

At the 1% level of significance, there is insufficient evidence to conclude that site and pottery type are not independent.    

Answer 1

(a) The Level of Significance is 0.01

The Hypothesis: Option 3

H0: Site Type and Pottery are Independent.

H1: Site Type and Pottery are Not Independent.

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(b) The Test Statistic:

The Expected value table has been given after the test.

The Calculation of the test statistic is as below

#	Observed	Expected	(O-E)	(O-E)2	(O-E)2/E
1	76	77.646	-1.646	2.709316	0.034893
2	85	87.506	-2.506	6.280036	0.071767
3	97	92.847	4.153	17.24741	0.185762
4	63	62.599	0.401	0.160801	0.002569
5	73	70.548	2.452	6.012304	0.085223
6	72	74.854	-2.854	8.145316	0.108816
7	50	48.755	1.245	1.550025	0.031792
8	55	54.946	0.054	0.002916	5.31E-05
9	57	58.299	-1.299	1.687401	0.028944
Total					0.550

= 0.550

Yes, All expected frequencies are greater than 5.

We use the Chi Square Distribution.

The Degrees of freedom = (r-1) * (c-1) = (3-1) * (3-1) = 2 * 2 = 4

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(c) The p value at = 0.550 is 0.9685

Therefore the range of the p values is Option 1: p value is > 0.100

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(d) Option 1: Since p value is > , we fail to reject the null hypothesis

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(e) Option 2: At the 1% level of significance, there is insufficient evidence to conclude that site and pottery type are not independent.

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