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   60   44   59   34   46   57
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 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.
H0: The distributions are the same.
H1: The distributions are the same.

(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?
uniform
chi-square
Student's t
normal

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 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, the evidence is sufficient to conclude that the distribution of observed outcomes for the die is different from the expected distribution of a fair die.
At the 1% level of significance, the evidence is insufficient to conclude that the distribution of observed outcomes for the die is different from the expected distribution of a fair die.

Answer 1

The statistical software output for this problem is :

Level of significance = 0.01

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

Chi square test statistics =

Yes

Chi square

Degrees of freedom = 5

0.050 < P-value < 0.100

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

At the 1% level of significance, the evidence is insufficient to conclude that the distribution of observed outcomes for the die is different from the expected distribution of a fair die.