Question

20. 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 X = the number of people in a randomly selected U.S. household and Y = the number of people in a randomly chosen U.S. family.
(a) Make histograms suitable for comparing the probability distributions of X and Y. Describe any differences that you observe.
(b) Find the mean for each random variable. Explain why this difference makes sense.
(c) Find the standard deviations of both X and Y. Explain why this difference makes sense.

Short answer

(a) Histograms of both distribution are skewed to the right.

(b) The mean for each random variable are $2.6$and $3.14$.

(c) The standard deviations of household and family are $1.4213$and $1.292$.

Step-by-step solution

Part (a) Step 1: Given information 

Given in the question that, the size of a house hold items in government data.

We have to make the histograms suitable for comparing the probability distributions of $X$and $Y$.

Part (a) Step 2: Graphical Representation

The histograms for the size of American households for the following types of households:

The histograms for Family Size in American Households as:

Part (a) Step 3: Explanation 

Because the top bar is to the left and the tail is to the right, both distributions are right-skewed. Because the tallest bars are at $2$, the center of both distributions is around two people. Both histograms have nearly the same spread.

The most notable difference in the histograms is that no American families with only one person exist, whereas American households with only one member do exist.

Part (b) Step 1: Given information 

Given in the question that, the table which shows the size of American households In government data. From the information we have to find the mean for each random variable.

Part (b) Step 2: Explanation 

We can compute the value of the mean for household by using the Ti-83 calculator as follow

Calculate the value of the mean for family using the Ti-83 calculator as follows:

Part (c) Step 1: Given information 

Given in the question that, the size of American households In government data.

We have to determine the standard deviations of both $X$and $Y$.

Part (c) Step 2: Explanation 

Compute the standard deviations for both distributions with the Ti-83 calculator as follows: