Question

The accompanying frequency distribution represents the square footage of a random sample of 500 houses that are owner occupied year round. Approximate the mean and standard deviation square footage.
Square footage Frequency 0- 499 500 999 13 1,000 1.499 33 1,500 1.999 115 2,000- 2.499 125 2,500 2.999 81 3,000- 3.499 3,500- 3.999 45 4,000 4.499 22 4,500 4.999 10

Answer 1

Answer:

$\bar X = \frac{\sum x_i f_i}{n} = \frac{1220750}{500}=2441.5$

$s^2= \frac{3408029125 -\frac{(1220750)^2}{500}}{500-1} =856849.7$

$s= \sqrt{856849.7}=925.662$

Step-by-step explanation:

For this case we can create the following table

Interval      Frequency (f)    Midpoint(xi)       xi *f      xi^2* f

0-499              9                       249.5            2245.5   560252.3

500-999         13                      749.5            9743.5    7302753

1000-1499      33                     1249.5          41233.5   51521258.25

1500-1999      115                    1749.5          201193.5  361986278.8

2000-2499     125                   2249.5         281187.5   632531281.3

2500-2999      81                    2749.5          222709.5 612339770.3

3000-3499      47                    3249.5         152726.5   496284761.8

3500-3999      45                    3749.5         168727.5    632643761.3

4000-4499      22                    4249.5         93489        397281505.5

4500-4999      10                     4749.5         47495        225577502.5

Total                500                                      1220750      3408029125

$\sum f_i = 500 , \sum x_i f_i = 1220750, \sum x^2_i f_i = 3408029125$

For this case we can calculate the sample mean with this formula:

$\bar X = \frac{\sum x_i f_i}{n} = \frac{1220750}{500}=2441.5$

And for the sample variance we can use the following formula:

$s^2= \frac{\sum x^2_i f_i - \frac{(\sum x_i f_i)^2}{n}}{n-1}$

And if we replace we got:

$s^2= \frac{3408029125 -\frac{(1220750)^2}{500}}{500-1} =856849.7$

And the deviation is just the square root of the sample variance and for this case is:

$s= \sqrt{856849.7}=925.662$