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	77	62	50	189
Cliff-Talus	75	67	71	213
Canyon Bench	96	64	66	226
Column Total	248	193	187	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 independent.
H₁: Site type and pottery are 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 not independent.
H₁: Site type and pottery 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?

Student's tbinomial    uniformnormalchi-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.)


(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

Answer:

Based on the given data, expected frequency for each pottery type was calculated:

Observed Frequency	Pottery Type
Site Type	Mesa Verda Black on White	McElmo Black on White	Mancos Black on White	Row Total
Mesa Top	77	62	50	189
Cliff-Talus	75	67	71	213
canyon Bench	96	64	66	226
Column Total	248	193	187	628

Expected Frequency	Pottery Type
Site Type	Mesa Verda Black on White	McElmo Black on White	Mancos Black on White	Row Total
Mesa Top	=248*189/628 = 75	=193*189/628 = 58	=187*189/628 = 56	189
Cliff-Talus	=248*213/628 = 84	=193*213/628 = 65	=187*213/628 = 63	213
canyon Bench	=248*226/628 = 89	=193*226/628 = 69	=187*226/628 = 67	226
Column Total	248	193	187	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?

• Level of significance is the likelihood that a statistical test will reject the data and hypothesis, despite the hypothesis actually being true. The predetermined level of significance for this example is 0.01. A statistically significant result in a hypothesis test is achieved when the p-value is less than the defined level of significance.

State the null and alternate hypotheses.

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

• The chi-square statistic is 3.932.

Are all the expected frequencies greater than 5?

• Yes  

What sampling distribution will you use?

• chi-square

What are the degrees of freedom?

• DF =(number of rows - 1)*(number of columns - 1) = (3-1)*(2-1) = 2

(c) Find or estimate the P-value of the sample test statistic. (Round your answer to three decimal places.)

• The p-value is 0.415. The result is not significant at p < 0.01.

(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.

(e) Interpret your conclusion in the context of the application.

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