Question

Fifty 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	5
3 up to 6	5
6 up to 9	10
9 up to 12	20
12 up to 15	10

a. Calculate the average vacancy rate.

b. Calculate the variance and the standard deviation for this sample. (Round intermediate calculations to at least 4 decimal places and final answers to 2 decimal places.)

Answer 1

Solution:

Given that

Class	Frequency	Mid value	fx	fx2
0 - 3	5	1.5	7.5	11.25
3 - 6	5	4.5	22.5	101.25
6 - 9	10	7.5	75	562.5
9 - 12	20	10.5	210	2205
12 - 15	10	13.5	135	1822.5
	n = 50		fx = 450	fx2 = 4702.5

Solution:

Given that

a ) The sample mean is

Mean   = (fx/ n)

= (450 / 50 )

= 9

The average vacancy rate = 9

b ) The  sample variance S²

  S² = (fx² ) - (( fx )² / n ) / 1 -n ) * h²

= ( 4702.5 ( ( 450 )² / 50 ) / 49

   = ( 4702.5 - 4050 / 50 ) / 49

= 652.5 / 49

= 13.3163

The  sample variance S² is 13.32

The standard deviation = sample variance

= 13.3163

= 3.6492

The standard deviation = 3.65