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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.)


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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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