Question

use a normal approximation to find the probability of the indicated number of voters. in this case, assume that 135 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 35 voted. the probability that fewer than 35 of 135 eligible voters voted is (round to four decimal places as needed.)

Explanation:

Step1: Identify parameters for normal - approximation

This is a binomial - to - normal approximation problem. Let \(n = 135\) (number of trials/sample size) and \(p=0.22\). The mean of the binomial distribution is \(\mu = np\) and the standard deviation is \(\sigma=\sqrt{np(1 - p)}\).
\(\mu=np=135\times0.22 = 29.7\)
\(\sigma=\sqrt{np(1 - p)}=\sqrt{135\times0.22\times(1 - 0.22)}=\sqrt{29.7\times0.78}=\sqrt{23.166}\approx4.8131\)

Step2: Apply continuity correction

For the binomial probability \(P(X\lt35)\), in normal approximation for a discrete variable, we find \(P(X\lt34.5)\) (continuity correction).
We standardize \(x = 34.5\) using the formula \(z=\frac{x-\mu}{\sigma}\).
\(z=\frac{34.5 - 29.7}{4.8131}=\frac{4.8}{4.8131}\approx0.9973\)

Step3: Find the probability using the standard normal table

We want to find \(P(Z\lt0.9973)\). Looking up the value in the standard - normal table (or using a calculator with a normal - distribution function), we get \(P(Z\lt0.9973)\approx0.8400\)

Answer:

\(0.8400\)