Chapter 8.3, Problem 8.56E

(a)

To determine

To calculate the probability that should Judy be suspicious that the company’s claim is not true, if only $15$ seeds actually germinate.

Answer to Problem 8.56E

  $0.1702$ .

Should not be suspicious.

Explanation of Solution

Given:

  $\begin{array}{l}n=20\\ p=0.85\end{array}$

Evaluate binomial at $k=16,17,18,19,20$ :

  $P(X=16)=C{}_{16}\times {(0.85)}^{16}\times {(1-0.85)}^{20-16}=0.1821$

  $P(X=17)=C{}_{17}\times {(0.85)}^{17}\times {(1-0.85)}^{20-17}=0.2428$

  $P(X=18)=C{}_{18}\times {(0.85)}^{18}\times {(1-0.85)}^{20-18}=0.2293$

  $P(X=19)=C{}_{19}\times {(0.85)}^{19}\times {(1-0.85)}^{20-19}=0.1368$

  $P(X=20)=C{}_{20}\times {(0.85)}^{20}\times {(1-0.85)}^{20-20}=0.0384$

Use addition rule:

  $\begin{array}{l}P(X>15)=P(X=16)+P(X=17)+P(X=18)+P(X=19)+P(X=20)\\ =0.1821+0.2428+0.2293+0.1368+0.0384\\ =0.8298\end{array}$

Use complement rule:

  $\begin{array}{l}P(X\le 15)=1-P(X>15)\\ =1-0.8298\\ =0.1702\end{array}$

Judy should not be suspicious that the company’s claim is untrue because the probability that $15$ seeds or less are germinated is not small.

(b)

To determine

To calculate the probability that should Judy be suspicious that the company’s claim is not true, if only $12$ seeds actually germinate.

Answer to Problem 8.56E

  $0.0059$

Judy should be suspicious.

Explanation of Solution

Given:

  $\begin{array}{l}n=20\\ p=0.85\end{array}$

Evaluate binomial at $k=13,14,15,16,17,18,19,20$ :

  $P(X=13)=C{}_{13}\times {(0.85)}^{13}\times {(1-0.85)}^{20-13}=0.0160$

  $P(X=14)=C{}_{14}\times {(0.85)}^{14}\times {(1-0.85)}^{20-14}=0.0454$

  $P(X=15)=C{}_{15}\times {(0.85)}^{15}\times {(1-0.85)}^{20-15}=0.1028$

  $P(X=16)=C{}_{16}\times {(0.85)}^{16}\times {(1-0.85)}^{20-16}=0.1821$

  $P(X=17)=C{}_{17}\times {(0.85)}^{17}\times {(1-0.85)}^{20-17}=0.2428$

  $P(X=18)=C{}_{18}\times {(0.85)}^{18}\times {(1-0.85)}^{20-18}=0.2293$

  $P(X=19)=C{}_{19}\times {(0.85)}^{19}\times {(1-0.85)}^{20-19}=0.1368$

  $P(X=20)=C{}_{20}\times {(0.85)}^{20}\times {(1-0.85)}^{20-20}=0.0384$

Use addition rule:

  $\begin{array}{l}P(X>12)=P(X=13)+P(X=14)+P(X=15)+P(X=16)+P(X=17)+P(X=18)+P(X=19)+P(X=20)\\ =0.0160+0.0454+0.1028+0.1821+0.2428+0.2293+0.1368+0.0384\\ =0.9941\end{array}$

Use complement rule:

  $\begin{array}{l}P(X\le 12)=1-P(X>12)\\ =1-0.9941\\ =0.0059\end{array}$

Judy should be suspicious that the company’s claim is untrue because the probability that $12$ seeds or less germinated is very small.