Chapter 10.1, Problem 5E

(a)

To determine

To find:The probability that the proportion of red jelly beans in the Child sample is less than or equal to the proportion of red jelly beans in the Adult sample.

Answer to Problem 5E

The probability that the proportion of red jelly beans in the Child sample is less than or equal to the proportion of red jelly beans in the Adult sample is $\mathrm{0.0212.}$

Explanation of Solution

Given information:

Proportion of red jelly beans in the child mix = 30%

Proportion of red jelly beans in the adult mix = 15%

Random samples of jelly beans taken from the Child Mix = 50

Random samples of jelly beans taken from the Adult Mix = 100

Calculation:

Red jelly beans in bags for children and $p_2$ is the actual proportion of red jelly beans in bags for adults, the sampling distribution of $p_1-p_2$ is Normal where $p_1=\frac{30}{100}=0.30,p_2=\frac{15}{100}=0.15$ because:

  $\begin{array}{cc}{n}_1{p}_1& =50\times 0.30=15\\ n_1\left(1-p_1\right)& =50\times 0.70=35\\ n_2{p}_2& =100\times 0.15=15\\ & \\ n_2\left(1-p_2\right)& =100\times 0.85=85\end{array}$

Are at least $10,$ the sampling distribution of ${\widehat{p}}_1-{\widehat{p}}_2$ is approximately Normal. Its mean is:

  $\begin{array}{cc}{\mu }_{\dot{\mu }-{\dot{p}}_2}& =p_1-p_2\\ & =0.30-0.15\\ & =0.15\end{array}$

The standard deviation is:

  $\begin{array}{cc}{\sigma }_{\dot{h}-{\widehat{h}}_2}& =\sqrt{\frac{p_1\left(1-p_1\right)}{n_1}+\frac{p_2\left(1-p_2\right)}{n_2}}\\ & =\sqrt{\frac{0.3\times (1-0.3)}{50}+\frac{0.15\times (1-0.15)}{100}}\\ & =0.0740\end{array}$

Hence,

  $p\left({\widehat{p}}_1-{\widehat{p}}_2\le 0\right)=p\left(z\le \frac{0-0.15}{0.0740}\right)$

  $=P(z\le -2.03)$ (From normal table)

  $=0.0212$

The probability that the proportion of red jelly beans in the Child sample is less than or equal to the proportion of red jelly beans in the Adult sample is $\mathrm{0.0212.}$

(b)

To determine

To Explain:The Child and Adult samples contain an equal proportion of red jelly beans. Based onresult in part (a)give reason to doubt the company's claim.

Answer to Problem 5E

Yes the company claim can be doubted.

Explanation of Solution

Yes, it is doubted the company's claim. From part (a), the probability that the proportion of red jelly beans in the Child sample is less than or equal to the proportion of red jelly beans in the Adult sample is 0.0212 . That is, there is only a $2\text{\%}$ chance of getting as few or fewer red jellybeans in the child sample than the adult sample if the company's claim is true. This isnot verylikely.