Question

In: Statistics and Probability

The following table shows site type and type of pottery for a random sample of 628...

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 79 57 53 189
Cliff-Talus 76 71 66 213
Canyon Bench 97 68 61 226
Column Total 252 196 180 628

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 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.

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

chi-square    

uniform

binomial

Student'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.)


(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 > ?, we fail to reject the null hypothesis.

Since the P-value > ?, we reject the null hypothesis.    

Since the P-value ? ?, we reject the null hypothesis.

Since the P-value ? ?, 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.    

Solutions

Expert Solution

(a) What is the level of significance?

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.

Let's use excel for calculation

The formulae used on the above excelsheet are as follows:

Are all the expected frequencies greater than 5 ?

Yes, all the expected frequencies greater than 5

What sampling distribution will you use?

normal

chi-square    

uniform

binomial

Student's t

The correct option is Chi-square

What are the degrees of freedom? four

(c) Find or estimate the P-value of the sample test statistic.

p-value = 0.603

(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis of independence?

Decision rule:

1) If p-value < level of significance (alpha) then we reject null hypothesis

2) If p-value > level of significance (alpha) then we fail to reject null hypothesis.

Here p value = 0.603 > 0.01 so we used 2nd rule.

That is we fail to reject null hypothesis

Since the P-value > ?, 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.  


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