Chapter 16, Problem 46E

(a)

To determine

To find: the probability that in this second box the overall weight reaches 60.5 lb which explicitly describes which notes all assumptions.

Answer to Problem 46E

0.074

Explanation of Solution

Given:

5-pound standard deviation = 0.1 lb

Bar mean = 10 lb

Bar standard deviation = 0.25 lb

20-pound weight mean = 20 lb.

20-pound standard deviation = 0.2 lb.

5-pound mean = 5 lb.

Formula used:

  $\text{z-score = }\frac{x-\mu }{\sigma }$

Calculation:

The weight of the four 5-pound weights (A) and the 2 20-pound weights (B) is needed to convey the box's total weight.

  $\text{Total weight = 4}\times \text{A+2}\times \text{B}$

The weight of the four 5-pound weights (A) and the 2 20-pound weights (B) is needed to convey the box's total weight.

  $\begin{array}{c}\text{Total weight, mean = 4}\times \text{A+2}\times \text{B}\\ \text{=4}\times \text{5+2}\times \text{20}\\ \text{=60 lb}\end{array}$

  $\begin{array}{c}\mathrm{var}\left(\text{Total weight}\right)\text{ = 4}\times \mathrm{var}{\left(\text{A}\right)}^2\text{+2}\times \mathrm{var}{\left(\text{B}\right)}^2\\ \text{=4}\times \text{var}{\left(0.1\right)}^2\text{+2}\times \text{var}{\left(0.2\right)}^2\\ \text{=0}\text{.12}\end{array}$

  $\begin{array}{c}\text{total weight, standard deviation=}\sqrt{\mathrm{var}\left(\text{Total weight}\right)}\\ =\sqrt{0.12}\\ =0.346\text{ lb}\end{array}$

Z-score is

  $\begin{array}{c}\text{z-score = }\frac{x-\mu }{\sigma }\\ =\frac{60.5-60}{0.346}\\ =1.45\end{array}$

The probability of a total weight of more than 60.5 pound are thus 0.074

(b)

To determine

To find: mean and default deviation of total company shipping expense for the start-up set.

Answer to Problem 46E

$0.186

Explanation of Solution

Given:

5-pound standard deviation = 0.1 lb

Bar mean = 10 lb

Bar standard deviation = 0.25 lb

20-pound weight mean = 20 lb.

20-pound standard deviation = 0.2 lb.

5-pound mean = 5 lb.

Calculation:

  $\text{Total cost =\$0}\text{.40}\times \text{A+\$0}\text{.50}\times \text{B+\$6}$

  $\begin{array}{c}\text{Total cost, mean =\$0}\text{.40}\times \text{A+\$0}\text{.50}\times \text{B+\$6}\\ \text{=\$0}\text{.40}\times 60\text{+\$0}\text{.50}\times 10\text{+\$6}\\ \text{=\$35}\end{array}$

Thus, the mean of the entire shipping cost is $35.

  $\begin{array}{c}\mathrm{var}\left(\text{Total cost}\right)\text{ =\$0}{\text{.40}}^2\times \mathrm{var}{\left(\text{A}\right)}^2\text{+\$0}{\text{.50}}^2\times \mathrm{var}{\left(\text{B}\right)}^2\\ \text{=\$0}{\text{.40}}^2\times \mathrm{var}{\left(0.346\right)}^2\text{+\$0}{\text{.50}}^2\times \mathrm{var}{\left(0.25\right)}^2\\ \text{=0}\text{.035}\end{array}$

  $\begin{array}{c}\text{Total Cost, standard deviation=}\sqrt{\mathrm{var}\left(\text{Total Cost}\right)}\\ =\sqrt{0.035}\\ =\$0.186\end{array}$

Hence, the standard deviation of the entire shipping cost is $0.186.

(c)

To determine

To find: the difference is likely to be more than one quarter of a pound.

Answer to Problem 46E

0.2877

Explanation of Solution

Given:

5-pound standard deviation = 0.1 lb

Bar mean = 10 lb

Bar standard deviation = 0.25 lb

20-pound weight mean = 20 lb.

20-pound standard deviation = 0.2 lb.

5-pound mean = 5 lb.

Formula used:

  $\text{z-score = }\frac{x-\mu }{\sigma }$

Calculation:

  $\text{difference,mean = 4}\times \text{A-B}$

  $\begin{array}{c}\text{Difference,mean = 4}\times \text{A-2}\times \text{B}\\ \text{=4}\times \text{5-20}\\ \text{=0 lb}\end{array}$

  $\begin{array}{c}\mathrm{var}\left(\text{Difference}\right){\text{ = 4}}^2\times \mathrm{var}{\left(\text{A}\right)}^2+\mathrm{var}{\left(\text{B}\right)}^2\\ {\text{=4}}^2\times \text{var}{\left(0.1\right)}^2+\mathrm{var}{\left(0.2\right)}^2\\ \text{=0}\text{.2}\end{array}$

  $\begin{array}{c}\text{Difference, standard deviation=}\sqrt{\mathrm{var}\left(\text{Difference}\right)}\\ =\sqrt{0.2}\\ =0.447\text{ lb}\end{array}$

Z-score is

  $\begin{array}{c}\text{z-score = }\frac{x-\mu }{\sigma }\\ =\frac{0.25-0}{0.447}\\ =0.56\end{array}$

Hence, the probability that the difference is lesser than a quarter of a pound is 0.2877