Question

I have a question from the Mathematical Statistics with Applications (7th edition) and the solution is provided by the website, but I am confused about part of the solution. The question reads:

The Bureau of the Census reports that the median family income for all families in the United States during the year 2003 was $43, 318. That is, half of all American families had incomes exceeding this amount, and half had incomes equal to or below this amount. Suppose that four families are surveyed and that each one reveals whether its income exceeded $43, 318 in 2003.

a) List the points in the sample space.

I found 15 points and the solution listed fifteen points, but then says that the are 16 points in the sample space and uses 16 for further calculations in the question.

What am I missing?

Thanks

Answer 1

Solution:

Given: four families are surveyed and that each one reveals whether its income exceeded $43, 318 in 2003

So responses could be Yes or No.

Part a) We have to list the points in the sample space.

Let Y = Yes and N = No.

Then we get following points in the sample space.

Responses	Outcomes						Frequency
All four say Yes	YYYY						1
Three Yes and One No	YYYN	YYNY	YNYY	NYYY			4
Two Yes and Two No	YYNN	YNYN	NYYN	NYNY	NNYY	YNNY	6
One Yes and Three Yes	YNNN	NYNN	NNYN	NNNY			4
All four say No	NNNN						1
							N = 16

So we can write in sample space as:

S : { YYYY , YYYN , YYNY , YNYY , NYYY , YYNN , YNYN   , NYYN   , NYNY , NNYY , YNNY ,

      YNNN , NYNN , NNYN , NNNY , NNNN    }