Question

Size of American households In government data, a household consists of all occupants of a dwelling unit, while a family consists of two or more persons who live together and are related by blood or marriage. So all families form households, but some households are not families. Here are the distributions of household size and family size in the United States:

Let H = the number of people in a randomly selected U.S. household and F= the number of people in a randomly chosen U.S. family.

(a) Here are histograms comparing the probability distributions of Hand F. Describe any differences that you observe.

(b) Find the expected value of each random variable. Explain why this difference makes sense.

(c) The standard deviations of the two random variables are ${\sigma }_H=1.421$and ${\sigma }_F=1.249$.Explain why this difference makes sense.

Short answer

Part (a) Both distributions are skewed to the right. The most typical number of people in a family and in a household is both 2.

Part (b) Household: 2.6 and Family: 3.14

Part (c) A household can only have one person, while a family must always have more than one person, hence there are more possible values for the number of people in a household (leading to a higher standard deviation).

Step-by-step solution

Part (a) Step 1. Given information.

The given information is:

Part (a) Step 2. Describe any differences that you observe. in the given histograms H and F.

The highest bars in the histograms are to the left, while a tail of lower bars is to the right, both distributions are skewed to the right.

The largest bar in both histograms is centered at 2, the most common number of people in a family and a household is both 2.

The histogram of families is narrower than the histogram of households, the spread of the number of people in a household is bigger than the spread of the number of people in a family.

There are no outliers in either distribution since there are no gaps in the histogram.

Part (b) Step 1. Find the expected value of each variable.

Mean of Household:

$$\begin{gathered} \mu =\sum xP\left(X=x\right) \\ =1\times 0.25+2\times 0.32+3\times 0.17+4\times 0.15+5\times 0.07+6\times 0.03+7\times 0.01 \\ =2.6 \end{gathered}$$

Mean of Family:

$$\begin{gathered} \mu =\sum xP\left(X=x\right) \\ =1\times 0+2\times 0.42+3\times 0.23+4\times 0.21+5\times 0.09+6\times 0.03+7\times 0.02 \\ =3.14 \end{gathered}$$

We can see that the Family has a greater expected value than the household which means all household has a minimum of one person and all families have a minimum of 2 persons.

Part (c) step 1. Differences in standard deviation.

The histogram of the household distribution is bigger, we inferred in part (a) that the spread of the home distribution was greater than the spread of the family distribution.

This is supported by standard deviations, which show that the household standard deviation is higher than the family standard deviation.

Furthermore, this makes sense because a household can only have one person, whereas a family must always have more than one person, and so there are more possible values for such a number of people in a household (leading to a higher standard deviation).