Question

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	9	15	17	41
Yearling	10	14	9	33
Adult	38	31	23	92
Column Total	57	60	49	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 not independent.
H₁: Age distribution and location are not independent.

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. (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?

Student's t binomial    

chi-square

uniform

normal

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 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.

Answer 1

Solution-: Let, A-Age and B- Location

(a) Hypothesis:

Two attribute Age distribution (A) and Location (B) are independent of each other.

Vs

Two attribute Age distribution (A) and Location (B) are not independent of each other. (or They are associated with each other)

Option (4) is correct.

(b)

The given frequencies table are observed frequencies

Age	Lamar District	Nez Perce District	Firehole District	Total
Calf	9	15	17	41
Yearling	10	14	9	33
Adult	38	31	23	92
Total	57	60	49	166

The corresponding expected frequencies are obtained using formula

for i,j=1,2,3

Note: are not to be rounded-off to the integers.

The given frequencies table are expected frequencies

Age	Lamar District	Nez Perce District	Firehole District
Calf	14.078	14.819	12.102
Yearling	11.331	11.928	9.741
Adult	31.590	33.253	27.157

Here,

Table for

5.754	15.183	23.880
8.825	16.432	8.315
45.710	28.900	19.480

The test statistic is

  

  

Yes, Here all frequencies are greater than 5

We will use sampling distribution is chi-square distribution.

Degree of freedom

(c) P-value:-     

By using MS-Excel Command "=CHIDIST(6.478,4)"

l.o.s.

(d) Option (1) Since the P-value > α, we fail to reject the null hypothesis is correct.

(e) Conclusion:- At the 5% level of significance, there is insufficient evidence to conclude that age distribution and location are not independent.