Question

The Fish and Game Department stocked a lake with fish in the following proportions: 30% catfish, 15% bass, 40% bluegill, and 15% pike. Five years later it sampled the lake to see if the distribution of fish had changed. It found that the 500 fish in the sample were distributed as follows.

Catfish	Bass	Bluegill	Pike
117	85	222	76

In the 5-year interval, did the distribution of fish change at the 0.05 level?

(A) What is the level of significance?

(B) State the null and alternate hypotheses.

H₀: The distributions are the same.
H₁: The distributions are the same.

H₀: The distributions are different.
H₁: The distributions are different.    

H₀: The distributions are different.
H₁: The distributions are the same.

H₀: The distributions are the same.
H₁: The distributions are different.

(C) 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.)

(D) Are all the expected frequencies greater than 5? (yes, no)

(E) What sampling distribution will you use? (binomial, Student's t , uniform, chi-square, normal)

(F) What are the degrees of freedom?

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

(H) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis that the population fits the specified distribution of categories?

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.

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

At the 5% level of significance, the evidence is insufficient to conclude that current fish distribution is different than that of five years ago.

At the 5% level of significance, the evidence is sufficient to conclude that current fish distribution is different than that of five years ago.    

Answer 1

(A) Level of significance = 0.05

(B) Null and Alternative hypothesis:

H0: The distributions are the same.
H1: The distributions are different.

(C) Value of Chi-square statistic = 11.027

(D) Yes, all the expected frequencies are greater than 5.

(E) Sampling distribution: Chi-square.

(F) Degrees of freedom = 3

(G) P-value = 0.012

(H) Since the P-value ≤ α, we reject the null hypothesis.

(I) At the 5% level of significance, the evidence is sufficient to conclude that current fish distribution is different than that of five years ago.

Calculations: