Question

Housing in San José How do rented housing units differ from units occupied by their owners? Here are the distributions of the number of rooms for owner-occupied units and renter-occupied units in San José, California:

Let X= the number of rooms in a randomly selected owner-occupied unit and Y = the number of rooms in a randomly chosen renter-occupied unit.

(a) Here are histograms comparing the probability distributions of X and Y. Describe any differences you observe.

(b) Find the expected number of rooms for both types of housing unit. Explain why this difference makes sense.

(c) The standard deviations of the two random variables are ${\sigma }_X=1.640$and ${\sigma }_Y=1.308$. Explain why this difference makes sense.

Short answer

Part (a) Both distributions have a minor rightward bias.

In owner-occupied homes, the most common number of rooms is 6, whereas, in renter-occupied units, the most common number of rooms is 4.

The number of rooms in owner-occupied units is more evenly distributed than the number of rooms in renter-occupied homes. There are no outliers in either distribution.

Part (b) Owned: 6.284 and Rented: 4.187

Part (c) The standard deviations confirm that the histogram of owner-occupied apartments is wider than the histogram of renter-occupied units.

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 X and Y.

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.

Because the tallest bar in the histogram for owner-occupied units is 6, the most common number of rooms in owner-occupied homes is 6. Because the highest bar in the histogram for rented-occupied apartments is 4, the most common number of rooms in renter-occupied units is four.

Because the width of the histogram for owner-occupied apartments is wider than the width of the histogram for renter-occupied units, the spread of the number of rooms in owner-occupied units is greater than the spread of the number of rooms in renter-occupied homes.

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 Owned:

$$\begin{gathered} \mu =\sum xP\left(X=x\right) \\ =1\times 0.003+2\times 0.002+3\times 0.023+4\times 0.104+5\times 0.210+6\times 0.224+ \\ 7\times 0.197+8\times 0.149+9\times 0.053+10\times 0.035 \\ =6.284 \end{gathered}$$

Mean of Rented:

$$\begin{gathered} \mu =\sum xP\left(X=x\right) \\ =1\times 0.008+2\times 0.027+3\times 0.287+4\times 0.363+5\times 0.164+6\times 0.093+ \\ 7\times 0.039+8\times 0.013+9\times 0.003+10\times 0.003 \\ =4.187 \end{gathered}$$

Because the width of the histogram for owner-occupied apartments is wider than the width of the histogram for renter-occupied units, the spread of the number of rooms in owner-occupied units is greater than the spread of the number of rooms in renter-occupied homes.

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

The histogram of owner-occupied units is larger, we inferred that the spread of the owner-occupied distribution was greater than the spread of the renter-occupied distribution in section (a).

The standard deviations reflect this, with the standard deviation of "owned" being higher than the standard deviation of "rented".

The standard deviations confirm that the histogram of owner-occupied apartments is wider than the histogram of renter-occupied units.