Question

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 between 25 and 30, inclusive, consumed alcohol before turning 21 years old. 0.1356 0.1527 0.1420 0.1252

Explanation:

Step1: Calculate the mean and standard deviation

The mean of a binomial distribution is \(\mu = np\), and the standard deviation is \(\sigma=\sqrt{np(1 - p)}\).
Given \(n = 50\) and \(p=0.68\), then \(\mu=np = 50\times0.68 = 34\) and \(\sigma=\sqrt{50\times0.68\times(1 - 0.68)}=\sqrt{50\times0.68\times0.32}=\sqrt{10.88}\approx3.3\)

Step2: Use the continuity correction

For \(X\) (number of people) between \(25\) and \(30\) inclusive, with continuity correction, we find \(P(24.5<X<30.5)\)

Step3: Standardize the values

The z - score formula is \(z=\frac{x-\mu}{\sigma}\)
For \(x = 24.5\), \(z_1=\frac{24.5 - 34}{3.3}=\frac{-9.5}{3.3}\approx - 2.88\)
For \(x = 30.5\), \(z_2=\frac{30.5 - 34}{3.3}=\frac{-3.5}{3.3}\approx - 1.06\)

Step4: Find the probability using the standard normal table

\(P(-2.88<Z<-1.06)=P(Z < - 1.06)-P(Z < - 2.88)\)
From the standard normal table, \(P(Z < - 1.06)=0.1446\), \(P(Z < - 2.88)=0.0020\)
\(P(-2.88<Z<-1.06)=0.1446 - 0.0020=0.1426\approx0.1420\)

Answer:

0.1420