A particular telephone number is used to receive both voice calls and fax messages. Suppose that 20% of the incoming calls involve fax messages, and consider a sample of 20 incoming calls. (Round your answers to three decimal places.)(a) What is the probability that at most 7 of the calls involve a fax message?(b) What is the probability that exactly 7 of the calls involve a fax message?(c) What is the probability that at least 7 of the calls involve a fax message?(d) What is the probability that more than 7 of the calls involve a fax message?

a) 0.968 = 96.8% probability that at most 7 of the calls involve a fax message

b) 0.055 = 5.5% probability that exactly 7 of the calls involve a fax message

c) 0.913 = 91.3% probability that at least 7 of the calls involve a fax message

d) 0.032 = 3.2% probability that more than 7 of the calls involve a fax message

Step-by-step explanation:

For each call, there are only two possible outcomes. Either there is a fax message, or there is not. The probability of a call being a fax is independent from other calls. So we use the binomial probability distribution to solve this question.

Binomial probability distribution

The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.

In which is the number of different combinations of x objects from a set of n elements, given by the following formula.

And p is the probability of X happening.

In this problem we have that:

(a) What is the probability that at most 7 of the calls involve a fax message?

This is .

So

In which

0.968 = 96.8% probability that at most 7 of the calls involve a fax message

(b) What is the probability that exactly 7 of the calls involve a fax message?

Rounded to three decimal places

0.055 = 5.5% probability that exactly 7 of the calls involve a fax message

(c) What is the probability that at least 7 of the calls involve a fax message?

Either less than 7 of the calls involve a fax message, or at least 7 does. The sum of the probabilities of these events is 1. So

We want . So

In which

0.913 = 91.3% probability that at least 7 of the calls involve a fax message

(d) What is the probability that more than 7 of the calls involve a fax message?

Either 7 or less calls involve a fax message, or at least 7 do. The sum of the probabilities of these events is decimal 1. So

We want P(X > 7).

From a, we have that . So

0.032 = 3.2% probability that more than 7 of the calls involve a fax message

a. the residual plot for model 1 has a random pattern and is a good fit for the data.