# Use a normal approximation to find the probability of the indicated number of voters. In this...

Use a normal approximation to find the probability of the indicated number of voters. In this case, assume that 156 eligible voters aged 18-24 are randomly selected. Suppose a previous study showed that among eligible voters aged 18-24,22 % of them voted.

Probability that fewer than 38 voted

The probability that fewer than 38 of 156 eligible voters voted is _______

(Round to four decimal places as needed.)

Solution

\begin{aligned} &\text { given : } n=156, p(\text { eligible voters })=0.22 \\ &n p=156 * 0.22=34.32 \geq 5 \\ &n(1-p)=156 *(1-0.22)=121.68 \geq 5 \end{aligned}

$$\therefore$$ Binomial random variable is approximately normal

$$\operatorname{Mean}(\mu)=n p=156 * 0.22=34.32$$

Standard deviation $$(\sigma)=\sqrt{n p(1-p)}=\sqrt{156 * 0.22 *(1-0.22)}=\sqrt{26.7696}$$ formula : $$Z=\frac{X-\mu}{\sigma}$$

$$P($$ fewer than 38) \Rightarrow P(X Use continuity correction.  \begin{aligned} &P(X<a)=P(X<a-0.5) \\ &\Rightarrow P(X<37.5) \\ &\Rightarrow P\left(\frac{X-\mu}{\sigma}<\frac{37.5-34.32}{\sqrt{26.7696}}\right) \\ &\Rightarrow P(Z<0.61) \end{aligned}  Refer to Z-table to find the probability or use the excel formula "=NORM.S.DIST(0.61, TRUE)" to find the probability.  \Rightarrow 0.7306  \(\therefore The probability that fewer than 38 of 156 eligible voters voted is $$0.7306$$

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