A recent survey by the American Automobile Association revealed that 80% of teen

Question

A recent survey by the American Automobile Association revealed that 80% of teenagers text while driving. You have been hired to do a safety presentation to a high school class of 100 teenage students. You will ask how many of them text while driving.

a) Explain why this procedure results in a binomial distribution.

b) State the values of n, p, and q for this distribution.

c) What is the mean number of teenagers in this class who text while driving?

d) What is the standard deviation of this distribution?

e) What is the probability that exactly 75 of these teenagers text while driving?

f) What is the probability that at most 75 of these teenagers text while driving?

g) What is the probability that at least 75 of these teenagers text while driving?

Explanation / Answer

a. A random variable is binomial if the following four conditions are met:

There are a fixed number of trials (n).

Each trial has two possible outcomes: success or failure.

The probability of success (call it p) is the same for each trial.

The trials are independent, meaning the outcome of one trial doesn't influence the outcome of any other trial.

As this problem has all this property it is binomial distribution.

b. n is 100 p is 0.8 and q is 0.2

c. Mean is np =100*0.8=80

d. Standard deviation is sqrt (npq)=sqrt(16)=4

e. P(x=75)=0.0439

f. P(x<=75)=0.1314

g. P(x>=75)=0.9125

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