Chapter 7.2, Problem 50E

To determine

To find: the equivalent to the probability of getting less than 50% democrats in a random sample of size 100.

Answer to Problem 50E

  $p\left(z<\frac{0.50-0.55}{\sqrt{\frac{0.55\times 0.45}{100}}}\right)$

Explanation of Solution

Given:

  $\begin{array}{l}p=0.55\\ \widehat{p}=0.50\\ n=100\end{array}$

Formula used:

  ${\sigma }_{\widehat{p}}=\sqrt{\frac{p\left(1-p\right)}{n}}$

Calculation:

  ${\mu }_{\widehat{p}}=p=0.55$

  $\begin{array}{c}{\sigma }_{\widehat{p}}=\sqrt{\frac{p\left(1-p\right)}{n}}\\ =\sqrt{\frac{0.55\left(1-0.55\right)}{100}}\\ =\sqrt{\frac{0.55\times 0.45}{100}}\end{array}$

  $\begin{array}{c}z=\frac{x-\mu }{\sigma }\\ =\frac{0.50-0.55}{\sqrt{\frac{0.55\times 0.45}{100}}}\end{array}$

Required probability that the sample proportion is less than 50%

  $p\left(\widehat{p}<50\%\right)=p\left(z<\frac{0.50-0.55}{\sqrt{\frac{0.55\times 0.45}{100}}}\right)$

Hence, the correct option is (b)