91影视

A Gallup Organization study showed that 20% of retail customers switched banks after their banks merged with another. After one year, a random sample of 250 retail customers was questioned if they changed their business from the First union to a different bank.

Let$\widehat{p}$ be the proportion of those customers who switched their business from First Union to a different bank.

The mean of the proportion is equal to the true binomial proportion p. That is,

$\begin{array}{c}E\left(\widehat{p}\right)={渭}_{\widehat{p}}\\ =p\\ =0.2\end{array}$

And the standard deviation of the proportion is defined as,

role="math" $\begin{array}{c}{蟽}_{\widehat{p}}=\sqrt{\frac{p\left(1鈭�p\right)}{n}}\\ =\sqrt{\frac{0.2\left(1鈭�0.2\right)}{250}}\\ =\sqrt{\frac{0.16}{250}}\\ =\sqrt{0.00064}\\ =0.0253\end{array}$

Thus, the mean and standard deviation of the proportion are 0.2and 0.0253 respectively.

Consider the interval$E\left(\widehat{p}\right)卤2{蟽}_{\widehat{p}}$

As there is obtained that$E\left(\widehat{p}\right)=0.2\text{鈥�}and\text{鈥�}{蟽}_{\widehat{p}}=0.0253$

So, the interval is,

$\begin{array}{c}E\left(\widehat{p}\right)卤2{蟽}_{\widehat{p}}=0.2卤2\left(0.0253\right)\\ =0.2卤0.0506\\ =\left(0.1494,0.2506\right)\end{array}$

Thus, the interval is (0.1494,0.2506).

Sample of size 250 were drawn repeatedly at a large number of times and determined the value of$\widehat{p}$ for each time.

In part b. there considered the interval (0.1494, 0.2506)

So, consider,

$\begin{array}{c}\mathrm{Pr}\left(0.1494鈮�\widehat{p}鈮�0.2506\right)=\mathrm{Pr}\left(\frac{0.1494鈭�p}{\sqrt{{蟽}_{\widehat{p}}}}鈮�\frac{\widehat{p}鈭�p}{\sqrt{{蟽}_{\widehat{p}}}}鈮�\frac{0.2506鈭�p}{\sqrt{{蟽}_{\widehat{p}}}}\right)\\ =\mathrm{Pr}\left(\frac{0.1494鈭�0.2}{0.0253}鈮�z鈮�\frac{0.2506鈭�0.2}{0.0253}\right)\\ =\mathrm{Pr}\left(\frac{鈭�0.0506}{0.0253}鈮�z鈮�\frac{0.0506}{0.0253}\right)\\ =\mathrm{Pr}\left(鈭�2鈮�z鈮�2\right)\\ =\mathrm{Pr}\left(z鈮�2\right)鈭�\mathrm{Pr}\left(z鈮�鈭�2\right)\\ =0.5+0.4772鈭�0.5+0.4772\\ =0.4772+0.4772\\ =0.9544\end{array}$

Therefore, the required proportion of the$\widehat{p}$ values that would fall within the interval of b is 0.9544.