Chapter PIV, Problem 26RE

(a)

To determine

To Calculate: In two consecutive days, the mean and standard deviation of the quantity researchers may spend.

Answer to Problem 26RE

Mean = $27

Standard Deviation = $9.89

Explanation of Solution

Given:

Mean = $13.50

Standard deviation $\left(\sigma \right)$ = $7

Formula used:

  $\begin{array}{c}E\left(X_1+X_2\right)=E\left(X_1\right)+E\left(X_2\right)\\ Var\left(X_1+X_2\right)=Var\left(X_1\right)+Var\left(X_2\right)\end{array}$

Calculation:

On a seven-day meal plan, a college student notes that the amount of money spent on food is with

From the given data.

Spending money on food every day = $13.5

Mean amount researcher spent money on two consecutive days

  $\begin{array}{c}E\left(X_1+X_2\right)=E\left(X_1\right)+E\left(X_2\right)\\ =13.5+13.5\\ =\$27\end{array}$

Mean amount spends on two days = $27

  $\begin{array}{c}Var\left(X_1+X_2\right)=Var\left(X_1\right)+Var\left(X_2\right)\\ ={\left(7\right)}^2+{\left(7\right)}^2\\ =49+49\\ =98\end{array}$

Therefore, standard deviation

  $\begin{array}{c}\left(\sigma \right)=\sqrt{98}\\ =\$9.89\end{array}$

Standard deviation for two days = $9.89.

(b)

To determine

To explain: that the assumption may be questioned by what assumption is made in order to obtain the standard deviation and any reasons.

Explanation of Solution

It may assume that the variables are independent by analysing the obtained results, so the reason for taking this statement is that the variable's covariance is zero.

(c)

To determine

To Calculate: the average cost of food per week and the standard deviation.

Answer to Problem 26RE

Standard deviation = $49

Standard deviation of the weekly food cost = $49

Explanation of Solution

We need to estimate the average weekly food cost and the standard deviation in this event.

  $\begin{array}{c}\text{Mean of the food on weekly costs = Mean }\left(\mu \right)\\ =E\left(7x\right)\\ =7\left(13.5\right)\\ =94.5\end{array}$

Mean of week food cost = 94.5

Variance is

  $\begin{array}{c}\mathrm{var}\left(7x\right)=49\left(49\right)\\ =2401\end{array}$

Standard deviation

  $\begin{array}{c}SD=\sigma \\ =\sqrt{2401}\\ =49\end{array}$

Standard deviation = $49

Standard deviation of the weekly food cost = $49

(d)

To determine

To explain: in analysis, including any assumptions made.

Explanation of Solution

By looking at the results, we can tell that in a week, he could spend less than $50.

Since they are 0.9 SDS below the variables of the principal assumptions, they are independent.