Question

6 multiple choice 1.43 points according to a recent survey, 68% of adult americans consumed alcohol before turning 21 years old. in a random sample of 50 adult americans, find the probability that less than 35 consumed alcohol before turning 21 years old. 0.7526 0.6317 0.8639 0.5530

Explanation:

Step1: Check conditions for normal approximation

We have a binomial distribution \(X\sim B(n = 50,p=0.68)\).
The mean \(\mu=np=50\times0.68 = 34\).
The standard deviation \(\sigma=\sqrt{np(1 - p)}=\sqrt{50\times0.68\times(1 - 0.68)}=\sqrt{50\times0.68\times0.32}=\sqrt{10.88}\approx3.3\).
Since \(np = 34\geq5\) and \(n(1 - p)=50\times0.32 = 16\geq5\), we can use the normal approximation \(X\sim N(\mu = 34,\sigma\approx3.3)\).

Step2: Apply continuity correction

We want \(P(X\lt35)\). Using the continuity correction for the binomial - to - normal approximation, we find \(P(X\lt35)=P(X\lt34.5)\) (for the normal approximation).
The z - score is \(z=\frac{x-\mu}{\sigma}\), where \(x = 34.5\), \(\mu = 34\), and \(\sigma\approx3.3\).
\(z=\frac{34.5 - 34}{3.3}=\frac{0.5}{3.3}\approx0.15\).

Step3: Find the probability using the standard normal table

We look up the value of \(P(Z\lt0.15)\) in the standard normal table.
From the standard normal table, \(P(Z\lt0.15)=0.5596 + 0.0536=0.7526\) (using the property that for \(z = 0.15\), \(P(Z\lt z)=\Phi(z)\) and \(\Phi(0.15)\) can be calculated as \(0.5+0.0596\) (the area from \(z = 0\) to \(z=0.15\))

Answer:

0.7526