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.

The given information is:

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.

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.