Family size. The Census Bureau gives this distribution for the number of people in American families in 2006:

Family size: 2 3 4 5 6 7 Proportion: 0.45 0.23 0.19 0.09 0.03 0.02 Chapter 20 Exercises 441

(Note: In this table, 7 actually represents families of size 7 or greater. But for purposes of this exercise, assume that it means only families of size exactly 7.)

(a) This is also the probability distribution (ignoring roundoff errors that make these proportions add to 1.01) for the size of a randomly chosen family. The expected value of this distribution is the average number of people in a family.

What is this expected value?

(b) Suppose you take a random sample of 1000 American families. How many of these families will be of size 2? Sizes 3 to 7?

(c) Based on your calculations in part (b), how many people are represented in your sample of 1000 families? (Hint: The number of individuals in your sample who live in families of size 7 is 7 times the number of families of size 7. Repeat this reasoning to determine the number of individuals in families of sizes 2 to 6. Add the results to get the total number of people represented in your sample.)

(d) Calculate the probability distribution for the family size lived in by individual people. Describe the shape of this distribution. What does this shape tell you about family structure?