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. it is expected that in a random sample of 400 adult smokers, 320 will have started smoking before turning
18.
b. it is expected that in a random sample of 400 adult smokers, 320 will have started smoking after turning 18
c. it is expected that in 50% of random samples of 400 adult smokers, 320 will have started smoking before
turning 18.

(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?
a. no, because 340 is greater than \\( \mu + 2 \sigma \\).
b. yes, because 340 is greater than \\( \mu + 2 \sigma \\).
c. no, because 340 is less than \\( \mu - 2 \sigma \\).
d. no, because 340 is between \\( \mu - 2 \sigma \\) and \\( \mu + 2 \sigma \\).
e. yes, because 340 is between \\( \mu - 2 \sigma \\) and \\( \mu + 2 \sigma \\).

Explanation:

Step1: Calculate the mean

The mean formula for a binomial distribution is $\mu = np$. Here, $n = 400$ and $p = 0.8$.
$$\mu=400\times0.8 = 320$$

Step2: Calculate the standard deviation

The standard deviation formula for a binomial distribution is $\sigma=\sqrt{np(1 - p)}$. Substitute $n = 400$ and $p = 0.8$.
$$\sigma=\sqrt{400\times0.8\times(1 - 0.8)}=\sqrt{400\times0.8\times0.2}=\sqrt{64}=8$$

Step3: Determine the range for usual values

Usual values lie within $\mu\pm2\sigma$. Calculate $\mu + 2\sigma=320+2\times8 = 336$ and $\mu - 2\sigma=320 - 2\times8=304$.
Since $340>336$ (i.e., $340>\mu + 2\sigma$)

Answer:

(a) Mean $\mu = 320$, Standard deviation $\sigma = 8$
(b) A. It is expected that in a random sample of 400 adult smokers, 320 will have started smoking before turning 18.
(c) B. Yes, because 340 is greater than $\mu + 2\sigma$.