Question

In: Statistics and Probability

The following table shows age distribution and location of a random sample of 166 buffalo in...

The following table shows age distribution and location of a random sample of 166 buffalo in a national park.

Age Lamar District Nez Perce District Firehole District Row Total
Calf 15 9 17 41
Yearling 10 8 15 33
Adult 31 24 37 92
Column Total 56 41 69 166

Use a chi-square test to determine if age distribution and location are independent at the 0.05 level of significance.

(a) What is the level of significance?


State the null and alternate hypotheses.

H0: Age distribution and location are independent.
H1: Age distribution and location are independent.H0: Age distribution and location are not independent.
H1: Age distribution and location are independent.    H0: Age distribution and location are independent.
H1: Age distribution and location are not independent.H0: Age distribution and location are not independent.
H1: Age distribution and location 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?

uniformchi-square    normalbinomialStudent'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.)

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 age distribution and location are not independent.At the 5% level of significance, there is insufficient evidence to conclude that age distribution and location are not independent.    

Solutions

Expert Solution

(a) What is the level of significance?

The level of significance is given as α = 0.05.

State the null and alternate hypotheses.

H0: Age distribution and location are independent.

H1: Age distribution and location are not independent.

(b) Find the value of the chi-square statistic for the sample.

Test statistic formula is given as below:

Chi square = ∑[(O – E)^2/E]

Where, O is observed frequencies and E is expected frequencies.

E = row total * column total / Grand total

Calculation tables for test statistic are given as below:

Observed Frequencies

Column variable

Row variable

Lamar

Nez

Firehole

Total

Calf

15

9

17

41

Yearling

10

8

15

33

Adult

31

24

37

92

Total

56

41

69

166

Expected Frequencies

Column variable

Row variable

Lamar

Nez

Firehole

Total

Calf

13.83133

10.12651

17.04217

41

Yearling

11.13253

8.150602

13.71687

33

Adult

31.03614

22.72289

38.24096

92

Total

56

41

69

166

(O - E)

1.168675

-1.12651

-0.04217

-1.13253

-0.1506

1.283133

-0.03614

1.277108

-1.24096

(O - E)^2/E

0.098747

0.125316

0.000104

0.115214

0.002783

0.12003

4.21E-05

0.071778

0.040271

Chi square = ∑[(O – E)^2/E] = 0.574285

Chi square test statistic = 0.574

Are all the expected frequencies greater than 5?

Yes

What sampling distribution will you use?

Chi square

What are the degrees of freedom?

We are given

Number of rows = r = 3

Number of columns = c = 3

Degrees of freedom = df = (r – 1)*(c – 1) = 2*2 = 4

Degrees of freedom = 4

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

P-value = 0.966

(by using Chi square table)

p-value > 0.100

(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 5% level of significance, there is insufficient evidence to conclude that age distribution and location are not independent.    


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