Question

Forty-nine 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 6 3 up to 6 6 6 up to 9 11 9 up to 12 21 12 up to 15 5 a. Calculate the average vacancy rate.b. Calculate the variance and the standard deviation for this sample.Variance: Standard deviation:

Answer 1

Answer:

$E(X) = \frac{431.5}{49}= 8.81$

$s^2 = \frac{4780.25- \frac{(431.5)^2}{49}}{48} = 20.425$

$s= \sqrt{20.425}= 4.52$

Step-by-step explanation:

For this case we can calculate all the questions with the following table:

Class   Midpoint(Xi)    fi      Xi*fi       Xi^2 *fi

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0-3          1.5                6        9           13.5

3-6          4.5               6        27          121.5

6-9          7.5               11        82.5      618.75

9-12         10.5             21       220.5    2315.25

12-25       18.5             5        18           1711.25

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Total                           49      431.5        4780.25

We can calculate the expected value with the following formula:

$E(X) = \frac{\sum_{i=1}^n X_i P(X_i)}{n}$

And if we replace we got:

$E(X) = \frac{431.5}{49}= 8.81$

We can calculate the sample variance with the following formula:

$s^2 =\frac{\sum X^2 f -\frac{(\sum xf)^2}{n} }{n-1}$

And replacing we got:

$s^2 = \frac{4780.25- \frac{(431.5)^2}{49}}{48} = 20.425$

And the deviation would be just the square root of the variance like this:

$s= \sqrt{20.425}= 4.52$