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 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.    

Solutions

Expert Solution

(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.

___________________________________

(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

________________________________

(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

_________________________________

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

_________________________________

(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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