Example 4.13 from Statistics, 4.3 Binomial Distribution (Optional)

Binomial Distribution (Optional): example 4.13

A store releases a 560-page art supply catalog. Eight of the pages feature signature artists. Suppose we randomly sample 100 pages. Let X = the number of pages that feature signature artists.

What values does x take on?
What is the probability distribution? Find the following probabilities:
1. the probability that two pages feature signature artists
2. the probability that at most six pages feature signature artists
3. the probability that more than three pages feature signature artists
Using the formulas, calculate the (i) mean and (ii) standard deviation.

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

1
Solution
1. x = 0, 1, 2, 3, 4, 5, 6, 7, 8
2. This is a binomial experiment since all three characteristics are met. Each page is a trial. Since we sample 100 pages, the number of trials is n = 100. For each page, there are two possible outcomes, features signature artists or does not feature signature artists. Since we are measuring the number of pages that feature signature artists, a page that features signature artists is defined as a success and a page that does not feature signature artists is defined as a failure. There are 8 out of 560 pages that feature signature artists. Therefore the probability of a success p=8560 and the probability of a failure q=1−p=1−8560=552560.
  
  Both p and q remain the same for each page. Therefore, X is a binomial random variable, and it can be written as X~B(100,8560).
  
  We can use a graphing calculator to answer Parts i to iii.
  1. P(x = 2) = binompdf $(100, \frac{8}{560}, 2)$ = .2466
  2. P(x ≤ 6) = binomcdf $(100, \frac{8}{560}, 6)$ = .9994
  3. P(x > 3) = 1 – P(x ≤ 3) = 1 – binomcdf $(100, \frac{8}{560}, 3)$ = 1 – .9443 = .0557
3. 1. mean = np = (100) $(\frac{8}{560})$ = $\frac{800}{560}$ ≈ 1.4286
  2. standard deviation = $\sqrt{n p q}$ = $\sqrt{(100) (\frac{8}{560}) (\frac{552}{560})}$ ≈ 1.1867

How did it go?

Solution

Solution