Question

Forty-two cities provided information on vacancy rates (in percent) for local apartments in the following frequency distribution.
Vacancy Rate (in percent) Frequency
0 up to 3 10
3 up to 6 9
6 up to 9 8
9 up to 12 7
12 up to 15 8
a. What is the average vacancy rate?
b. What is the variance?
c. What is the Standard Deviation?

Answer 1

Answer:

$(a)\ \bar x = 7.07$

$(b)\ \sigma^2 = 18.67$

$(c)\ \sigma = 4.32$

Step-by-step explanation:

Given

$\begin{array}{cc}{Rate (\%)} & {f} & {0-3} & {10} & {3-6} & {9} & {6-9} & {8} & {9-12} & {7} & {12-15} & {8} \ \end{array}$

Solving (a): The average vacancy rate

First, we calculate the class midpoint (x). This is the mean of the class boundaries.

$x = \frac{0+3}{2} = 1.5$     $x = \frac{3+6}{2} = 4.5$    $x = \frac{6+9}{2} = 7.5$

When done for the whole class, the frequency table becomes:

$\begin{array}{ccc}{Rate (\%)} & {f} & {x} & {0-3} & {10} &{1.5}& {3-6} & {9} &{4.5}& {6-9} & {8} & {7.5}&{9-12} & {7} &{10.5}& {12-15} & {8} &{13.5}\ \end{array}$

The average rate is:

$\bar x = \frac{\sum fx}{\sum f}$

$\bar x = \frac{10*1.5 + 9*4.5+8*7.5+7*10.5+8*13.5}{10+9+8+7+8}$

$\bar x = \frac{297}{42}$

$\bar x = 7.07$

Solving (b): The variance

This is calculated as:

$\sigma^2 = \frac{\sum f(x -\bar x)^2}{\sum f}$

$\sigma^2 = \frac{10(1.5-7.07)^2 + 9(4.5-7.07)^2+8(7.5-7.07)^2+7(10.5-7.07)^2+8(13.5-7.07)^2}{10+9+8+7+8}$

$\sigma^2 = \frac{784.2858}{42}$

$\sigma^2 = 18.67$

Solving (c): The standard deviation.

This is calculated as:

$\sigma = \sqrt{\sigma^2$

$\sigma = \sqrt{18.67$

$\sigma = 4.32$