Chapter 10.1, Problem 10.9E

(a)

To determine

To Explain: the appropriate hypotheses for a significance test of the company’s claim.

Answer to Problem 10.9E

  $\begin{array}{c}{H}_0:p_1=p_2=p_3=p_4=20\%=0.20\\ H_1:{\text{At least one of the p}}_i\text{'}\text{s is different}\end{array}$

Explanation of Solution

Given:

  $\begin{array}{c}n=60\\ \alpha =0.05/0.01\end{array}$

The null hypothesis statement is the population proportions are equal to the given proportions

  $H_0:p_1=p_2=p_3=p_4=20\%=0.20$

The alternative hypothesis statement is the opposite of the null hypothesis:

  $H_1:{\text{At least one of the p}}_i\text{'}\text{s is different}\text{.}$

(b)

To determine

To Calculate: the expected counts for a bag of skittles with 60 candies.

Answer to Problem 10.9E

  ${\text{E}}_1={\text{E}}_2=E_3=E_4=E_5=12$

Explanation of Solution

Given:

  $\begin{array}{c}n=60\\ \alpha =0.05/0.01\end{array}$

Formula used:

  $E=np$

Calculation:

The expected frequencies $E$ are.

  $\begin{array}{c}{\text{E}}_1=n{p}_1=60\times 0.20=12\\ {\text{E}}_2=n{p}_2=60\times 0.20=12\\ E_3=n{p}_3=60\times 0.20=12\\ E_4=n{p}_4=60\times 0.20=12\\ E_5=n{p}_5=60\times 0.20=12\end{array}$

(c)

To determine

To Explain: a chi-square statistic would need to get an order to have significant evidence against the company’s claim at the 0.05 level.

Answer to Problem 10.9E

  ${\chi }_{\alpha }^2=9.49$

Explanation of Solution

Given:

  $\alpha =0.05$

The critical value is the given in the row with $df=c-1=5-1=4$ and in the column corresponding $\alpha =0.05$ of the chi-square distribution table:

  ${\chi }_{\alpha }^2=9.49$

The rejection region than contains all values which are greater than 9.49, therefore significance evidence against the company’s claim if the ${\chi }^2$ -statistic is greater than 9.49.

(d)

To determine

To Explain: a chi-square statistic would need to get an order to have significant evidence against the company’s claim at the 0.01 level.

Answer to Problem 10.9E

  ${\chi }_{\alpha }^2=13.28$

Explanation of Solution

Given:

  $\alpha =0.01$

The critical value is the given in the row with $df=c-1=5-1=4$ and in the column with $\alpha =0.01$ of the chi-square distribution table

  ${\chi }_{\alpha }^2=13.28$

The rejection region then contains all values which are greater than 13.28, therefore significance evidence against the company’s claim if the ${\chi }^2$ -statistic is greater than 13.28