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}= \\)

Explanation:

Step1: Identify the distribution parameters

This is a binomial distribution problem. For a binomial distribution \(X\sim B(n,p)\), where \(n = 400\) (number of trials) and \(p=0.8\) (probability of success). The mean of a binomial distribution is \(\mu_{X}=np\).

$$ \mu_{X}=400\times0.8 $$

Step2: Calculate the mean

$$ \mu_{X}=320 $$

The standard deviation of a binomial distribution is \(\sigma_{X}=\sqrt{np(1 - p)}\). First, find \(1-p=1 - 0.8=0.2\). Then

$$ \sigma_{X}=\sqrt{400\times0.8\times0.2}=\sqrt{64}=8 $$

Answer:

\(\mu_{X} = 320\)