Question

according to an almanac, 80% of adult smokers started smoking before turning 18 years old
(a) compute the mean and standard deviation of the random variable x, the number of smokers who started before 18
in 400 trials of the probability experiment
(b) interpret the mean
(c) would it be unusual to observe 340 smokers who started smoking before turning 18 years old in a random sample
of 400 adult smokers? why?
(a) \\( \mu _ { x } = 320 \\)
\\( \sigma _ { x } = \square \\) (round to the nearest tenth as needed )

Explanation:

Step1: Recall the formula for standard deviation of a binomial distribution

For a binomial distribution $X\sim B(n,p)$, the standard deviation is $\sigma_X=\sqrt{np(1 - p)}$, where $n = 400$ and $p=0.8$.

Step2: Substitute the values into the formula

First, calculate $1-p=1 - 0.8=0.2$. Then $\sigma_X=\sqrt{400\times0.8\times0.2}$.

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Answer:

$\sigma_X = 8.0$