20.21 Household size. The Census Bureau gives this distribution for the number of people in American households in 2011:
| Family size: | 1 | 2 | 3 | 4 |
| Proportion: | 0.28 | 0.34 | 0.16 | 0.13 |
| Family size: | 5 | 6 | 7 | |
| Proportion: | 0.06 | 0.02 | 0.01 |
(Note: In this table, 7 actually represents households of size 7 or greater. But for purposes of this exercise, assume that it means only households of size exactly 7.)
(a) This is also the probability distribution for the size of a randomly chosen households. The expected value of this distribution is the average number of people in a household. What is this expected value?
(b) Suppose you take a random sample of 1000 American households. About how many of these households 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 households? (Hint: The number of individuals in your sample who live in households of size 7 is 7 times the number of households of size 7. Repeat this reasoning to determine the number of individuals in households 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 household size lived in by individual people. Describe the shape of this distribution. What does this shape tell you about household structure?