Question

In a congressional district, 55% of the registered voters are Democrats. Which of the following is equivalent to the probability of getting less than 50% Democrats in a random sample of size 100?

a. P(z<0.50−0.55100)

b. P(z<0.50−0.550.55(0.45)100)

c. P(z<0.55−0.500.55(0.45)100)

d. P(z<0.50−0.55100(0.55)(0.45))

e. P(z<0.55−0.50100(0.55)(0.45))

Short answer

The correct option is (b) $p\left(z<\frac{0.50-0.55}{\sqrt{{\displaystyle \frac{0.55\times 0.45}{100}}}}\right)$

Step-by-step solution

Given information

$$\begin{gathered} p=0.55 \\ \hat{p}=0.50 \\ n=100 \end{gathered}$$

Concept

${\sigma }_{\hat{p}}=\sqrt{\frac{p(1-p)}{n}}$

Calculation

$$\begin{gathered} {\mu }_{\hat{p}}=p=0.55 \\ {\sigma }_{\hat{p}}=\sqrt{\frac{p(1-p)}{n}} \\ =\sqrt{\frac{0.55(1-0.55)}{100}} \\ =\sqrt{\frac{0.55\times 0.45}{100}} \\ z=\frac{x-\mu }{\sigma } \\ =\frac{0.50-0.55}{\sqrt{{\displaystyle \frac{0.55\times 0.45}{100}}}} \end{gathered}$$

The sample proportion must be less than $50\%$ with a probability of less than $50\%$

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

Hence, the correct option is (b)