Chapter 26, Problem 4E

a)

To determine

To find the expected value for each color.

Explanation of Solution

Formula:

Y13y $E=np$

Calculation:

  

b)

To determine

To explain which test would be appropriate.

Answer to Problem 4E

Chi-square goodness of fit.

Explanation of Solution

Given:

  

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 4E

  $\begin{array}{l}{H}_0:p_1=p_2=0.20,p_3=p_4=p_5=0.10,p_6=0.3\\ H_a:Atleastoneofthe{p}_i’sdifferent\end{array}$

Explanation of Solution

Given:

  

The null and alternative hypotheses are:

  $\begin{array}{l}{H}_0:p_1=p_2=0.20,p_3=p_4=p_5=0.10,p_6=0.3\\ H_a:Atleastoneofthe{p}_i’sdifferent\end{array}$

d)

To determine

To check the conditions.

Answer to Problem 4E

All the conditions satisfied.

Explanation of Solution

Given:

  

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 M&M’s are random so condition of independent is satisfied.

The expected frequencies areat least 5 so all the conditions satisfied.

e)

To determine

To find the degrees of freedom.

Answer to Problem 4E

The degrees of freedom = 5

Explanation of Solution

Given:

  

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 4E

  ${\chi }^2$ = 9.31 and P-value = 0.0972

Explanation of Solution

Given:

  

Using excel,

  

So,

  ${\chi }^2$ = 9.31

P-value = 0.0972

g)

To determine

To state the conclusion.

Answer to Problem 4E

There is not sufficient evidence to say that the distribution is other than the specified proportions.

Explanation of Solution

Given:

  

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

Conclusion: There is not sufficient evidence to say that the distribution is other than the specified proportions.