91Ó°ÊÓ

X denotes the amount of soft drink.

X follows a uniform distribution.

Since the amount of soft drinks can be measured, so the amount dispensed by the beverage machine is a continuous random variable.

When dispense is 7 ounces, the graph of the frequency function X is as follows

Let,

$\begin{array}{l}c=6.5\\ d=7.5\end{array}$

Since X follows a uniform distribution,the mean is

$\begin{array}{c}\mathrm{μ}=\frac{c+d}{2}\\ =\frac{6.5+7.5}{2}\\ =7\end{array}$

Hence, the mean is $\mathrm{μ}=7$.

The standard deviation is

$\begin{array}{c}\mathrm{σ}=\frac{d−c}{\sqrt{12}}\\ =\frac{7.5−6.5}{\sqrt{12}}\\ =0.2886\end{array}$

Hence, the standard deviation is$\mathrm{σ}=0.2886$.

The interval $\mathrm{μ}Â\pm 2\mathrm{σ}$is

$\begin{array}{c}\mathrm{μ}Â\pm 2\mathrm{σ}=7Â\pm (2×0.2886)\\ =7Â\pm 0.5772\\ =(6.4228,7.5772)\end{array}$

Hence, the interval $\mathrm{μ}Â\pm 2\mathrm{σ}$ is$(6.4228,7.5772)$.

The mean and the interval $\mathrm{μ}Â\pm 2\mathrm{σ}$are located below the graph,

The p.d.f of X is

$\begin{array}{c}f\left(x\right)=\frac{1}{d−c};c≤x≤d\\ =\frac{1}{7.5−6.5};6.5≤x≤7.5\\ =1\end{array}$

$\begin{array}{c}P(xâ‰\yen 7)=1−P(x<7)\\ =1−\underset{6.5}{\overset{7}{∫}}f\left(x\right)dx\\ =1−\underset{6.5}{\overset{7}{∫}}1dx\\ =1−(7−6.5)\\ =1−0.5\\ =0.5\end{array}$

Since,

$\begin{array}{c}f\left(x\right)=\frac{1}{7.5−6.5};6.5≤x≤7.5\\ =1\end{array}$

So$P(x<6)=0$,

$\begin{array}{c}P(6.5≤x≤7.25)=\underset{6.5}{\overset{7.25}{∫}}f\left(x\right)dx\\ =\underset{6.5}{\overset{7.25}{∫}}1dx\\ =(7.25−6.5)\\ =\text{0.75}\end{array}$

Hence, $P(6.5≤x≤7.25)=\text{0.75}$.

The probability that six bottles will contain more than 7.25 ounces of beverage is .

$P(x=6)$

$\begin{array}{c}P(x>7.25)=\underset{7.25}{\overset{7.5}{∫}}f\left(x\right)dx\\ =\underset{7.25}{\overset{7.5}{∫}}1dx\\ =(7.5−7.25)\\ =0.25\end{array}$

Since,$x>7.25$ is a binomial vitiate $B(0.25,6)$.

So,

$\begin{array}{c}P(x=6)=\left(\begin{array}{c}6\\ 6\end{array}\right){\left(0.25\right)}^6{(1−0.25)}^{6−6}\\ ={\left(0.25\right)}^6\\ =0.00024\end{array}$

Hence, the probability that six bottles will contain more than 7.25 ounces of beverage is 0.00024.