Question

A gambler complained about the dice. They seemed to be loaded! The dice were taken off the table and tested one at a time. One die was rolled 300 times and the following frequencies were recorded.

Outcome	1	2	3	4	5	6
Observed frequency O	63	46	60	36	43	52

Do these data indicate that the die is unbalanced? Use a 1% level of significance. Hint: If the die is balanced, all outcomes should have the same expected frequency.

(a) What is the level of significance?


State the null and alternate hypotheses.

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

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

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

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

(b) Find the value of the chi-square statistic for the sample. (Round your answer to three decimal places.)


Are all the expected frequencies greater than 5?

Yes

No    

What sampling distribution will you use?

Student's t

uniform    

binomial

normal

chi-square

What are the degrees of freedom?


(c) Estimate the P-value of the sample test statistic.

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 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.

(e) 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

Solution:

a.) level of significance (α)=0.01

Null and Alternate hypotheses.

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

b.)

Probability of each event=1/6

n=300

Test Statistic(X²)= 10.680

Are all the expected frequencies greater than 5?

Ans:Yes
What sampling distribution will you use?

Ans: Chi Square

c.)

Degree of freedom=6-1=5

p-Value=0.0581 (use excel function:=CHIDIST(10.68,5))

ans: 0.050 < P-value < 0.100  

d.) P-value >significance level(α=0.01), Fail to reject Null hypothesis.

Ans:Since the P-value > α, we fail to reject the null hypothesis.

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