Question

according to an almanac, 70% 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 smoking before 18 based on a random sample of 200 adult smokers.
(b) interpret the mean.
(a) \\( \mu _ { x } = 140 \\)
\\( \sigma _ { x } = 6.5 \\) (round to the nearest tenth as needed.)
(b) what is the correct interpretation of the mean?
a. it is expected that in 50% of random samples of 200 adult smokers, 140 will have started smoking before turning 18.
b. it is expected that in a random sample of 200 adult smokers, 140 will have started smoking before turning 18.
c. it is expected that in 50% of random samples of 200 adult smokers, 140 will have started smoking after turning 18.
d. it is expected that in a random sample of 200 adult smokers, 140 will have started smoking after turning 18.

Explanation:

The mean of a binomial distribution \(X\) (where \(X\) represents the number of smokers who started before 18) with parameters \(n = 200\) (sample size) and \(p=0.7\) (probability of success - starting to smoke before 18) is \(\mu = np\). The standard deviation is \(\sigma=\sqrt{np(1 - p)}\). For part (b), the mean of a random variable in this context represents the expected value in a single sample.

Answer:

For part (a), \(\mu_X=np = 200\times0.7=140\), \(\sigma_X=\sqrt{np(1 - p)}=\sqrt{200\times0.7\times(1 - 0.7)}=\sqrt{42}\approx6.5\).

For part (b), the mean of a random variable \(X\) (in a binomial - like situation here) is the expected value of \(X\) in a single sample.

• Option A is incorrect because the mean is not about 50% of samples.
• Option B is correct. Since \(\mu_X = 140\), it is expected that in a random sample of \(n = 200\) adult smokers, \(140\) will have started smoking before turning 18.
• Option C is incorrect as it refers to starting after 18 (opposite of what \(X\) represents).
• Option D is incorrect as it refers to starting after 18 (opposite of what \(X\) represents).

So, for part (b) the answer is B.