Question

 According to an almanac, 80% of adult smokers started smoking before turning 18 years old. When technology is used, use the Tech Help button for further assistance.

 (a) Compute the mean and standard deviation of the random variable X, the number of smokers who started before 18 in 200 trials of the probability experiment.

 (b) Interpret the mean.

 (c) Would it be unusual to observe 170 smokers who started smoking before turning 18 years old in a random sample of 200 adult smokers? Why?

Answer 1

Concepts and reason

A binomial experiment is a discrete probability experiment that satisfies the following conditions. The experiment is repeated for a fixed number of trails, each trail is independent of the other trails. There are two possible outcomes for each trail. The outcomes are classified as success (S) or as a failure (F).

The probability of success, p remains same for each trail. The random variable X represents the number of successes in n independent trails of the experiment.

Fundamentals

The probability of success is known and number of trails is fixed. So, apply binomial distribution.

Let the random variable X follows a binomial distribution with parameters n and p.

The probability mass function (PMF) of binomial distribution can be defined as,

The mean and standard deviation of the random variable X is,

(a)

Let the random variable X represents the number of smokers who started before 18in 200 trails of the probability experiment.

Let represent the number of randomly selected adult smokers.

Let p=0.80 represent the proportion of adult smokers started smoking before turning 18 years old.

The possible outcomes are “Before turning 18 years old” or “After turning 18 years old.”

The experiment is repeated n times (countable finite).

The mean of the random variable X is,

The standard deviation of the random variable X is,

(b)

The expected value of the random variable X is considered as the mean of the random variable.

Hence, it is expected that in a random sample of 200 adult smokers, 160 will have started smoking before turning 18.

(c)

In part (a), the mean and standard deviation for the binomial variable are, and , respectively.

The 2 sigma limits for the smokers who started smoking before 18 are calculated as follows:

The value 170 lies between the 2 sigma limits.

Ans: Part a

The mean and standard deviation of the random variable X are and .

Part b

It is expected that in a random sample of 200 adult smokers, 160 will have started smoking before turning 18.

Part c

No, because 170 is lies between and.