Chapter 26, Problem 3E

a)

To determine

To find expected value.

Answer to Problem 3E

Expected value is 10.

Explanation of Solution

Given:

Face	Count
1	11
2	7
3	9
4	15
5	12
6	6

Formula:

  $E=np$

Calculation:

The die roll 60 times. Therefore, sample size would be n = 60.

There is equal chance to get a face value when die is rolled. Therefore,

  $p=\frac{1}{6}$

So, expected value is,

  $E=60\times \frac{1}{6}=10$

b)

To determine

To explain which test would be appropriate.

Answer to Problem 3E

Chi-square goodness of fit.

Explanation of Solution

Given:

Face	Count
1	11
2	7
3	9
4	15
5	12
6	6

The researcher wants to test whether the distribution is uniform. The test involves only one variable so, Chi-square goodness of fit would be appropriate to use.

c)

To determine

To state null and alternative hypotheses.

Answer to Problem 3E

  $\begin{array}{l}{H}_0:p_1=p_2=\mathrm{...}{p}_6=\frac{1}{6}\\ H_a:Atleastoneofthe{p}_i’sdifferent\end{array}$

Explanation of Solution

Given:

Face	Count
1	11
2	7
3	9
4	15
5	12
6	6

The null and alternative hypotheses:

  $\begin{array}{l}{H}_0:p_1=p_2=\mathrm{...}{p}_6=\frac{1}{6}\\ H_a:Atleastoneofthe{p}_i’sdifferent\end{array}$

d)

To determine

To check the conditions.

Answer to Problem 3E

All the conditions satisfied.

Explanation of Solution

Given:

Face	Count
1	11
2	7
3	9
4	15
5	12
6	6

There are three main conditions of Chi-square test:

Counts, Independent and expected frequencies are at least 5.

The data has given counts so the condition of counts satisfied.

The rolls are random so condition of independent is satisfied.

The expected frequency is 10 which is greater than 5 so all the conditions satisfied.

e)

To determine

To find the degrees of freedom.

Answer to Problem 3E

The degrees of freedom are 5

Explanation of Solution

Given:

Face	Count
1	11
2	7
3	9
4	15
5	12
6	6

The degrees of freedom are:

  $df=c-1=6-1=5$

f)

To determine

To find the ${\chi }^2$ and P-value.

Answer to Problem 3E

  ${\chi }^2$ = 5.60 and P-value = 0.3471.

Explanation of Solution

Given:

Face	Count
1	11
2	7
3	9
4	15
5	12
6	6

Using excel,

  

So,

  ${\chi }^2$ = 5.60

P-value = 0.3471

g)

To determine

To state the conclusion.

Answer to Problem 3E

There is not sufficient evidence to reject the claim that the die is fair.

Explanation of Solution

Given:

Face	Count
1	11
2	7
3	9
4	15
5	12
6	6

Decision: The p-value> 0.05, fail to reject H0.

Conclusion: There is not sufficient evidence to reject the claim that the die is fair.