Question

The following table gives the frequency distribution of the daily commuting times (in minutes) from home to work for a sample of 25 employees of a company.

Daily Commuting Time (mins)	Number of Employees f	Relative Frequency	Cumulative Frequency	Midpoint x	xf	(x - x̄)2
0 to less than 10	4	4/25	4	5	20	268.96
10 to less than 20	9	9/25	13	15	135	40.96
20 to less than 30	6	6/25	19	25	150	12.96
30 to less than 40	4	4/25	23	35	140	184.96
40 to less than 50	2	2/25	25	45	90	556.96
	25				535	1064.80

1. Construct a relative frequency distribution table for this data.

Relative frequency = frequency divided by total cumulative frequency

∑f = 4+9+6+4+2=25

Calculating ∑f: 4+9=13, 13+6=19, 19+4=23, 23+2=25

1. What proportion of employees has daily commute time less than 30 minutes?

4+9+6=19

1. Calculate the median daily commute time of employees.

n = 5, 5/2 = 2.5, (∑f)₁ = 4, L₁ = 10, C = 20 – 10 = 10, ƒ median = 9

The median class is 10 to less than 20

Me = 10 + (2.5-49)10 = 10 + (-1.5/9) 10 = 10 + (-1/6) 10 = 10 + -1 2/3 = 8 1/3

1. Calculate the median daily commute time of employees.

n= 5, 5/2 = 2.5, (∑f)₁ = 4, L₁ = 10, C = 20 – 10 = 10, ƒ median = 9

The median class is 10 to less than 20

Me = 10 + (2.5-4/9) 10 = 10 + (-1.5/9) 10 = 10 + (-1/6) 10 = 10 + -1 2/3 = 8 1/3

1. Calculate the mean daily commute time of employees.

First calculate the midpoint: (0+10)/2=5, (10+20)/2=15, (20+30)/2=25, (30+40)/2=35, (40+50)/2=45

Then the sum of xf: 4*5=20, 9*15=135, 6*25=150, 4*35=140, 2*45=90

Therefore, 20+135+150+140+90=535

Mean = 535/25=21.4

1. Calculate the variance for the daily commute time of employees.

Sample Variance = ∑(x-x̄n-1 )²

x̄ = 21.4

5-21.4= (-16.4)² = 268.96; 15-21.4= (-6.4)² = 40.96; 25-21.4= (3.6)² = 12.96;

35-21.4= (13.6)² = 184.96; 45-21.4= (23.6)² = 556.96

∴ ∑(x - x̄)2 = 286.96+40.96+12.96+184.96+556.96 = 1064.80

n =5; 5-1=4

∴ 1064.80/4 = 266.20

n 2 Me L C wmedian n 2 Me L C wmedian

Answer 1

(a)

Relative frequency = frequency divided by total frequency

Total frequency= ∑f = 4+9+6+4+2=25

So Respective Relative Frequencies are 4/25, 9/25, 6/25, 4/25, 2/25

(b)

Proportion of employees = number of employees / total employees

Number of employees less than 30 are 4+9+6=19

Total employees are 25

So Proportion = 19 / 25 = 0.76

NOW,

r 25 x-A Class Frequency (fMid value (x) (2) fd (5) (2) x (4)(6) = (5) x (4)| (7) cf h 10 f- a () 25, h 10 (4) (3) A 2 -8 0 10 4 16 4 10 20 9 15 -1 13 20 30 25-A C 19 30 40 4 35 1 4 4 23 40-50 2 45 2 4 8 25 Σr. d--9|Σr d-37 n = 25 et

(c) MEDIAN

To find Median Class
= value of (n/2)th observation

= value of (25/2)th observation

= value of 12th observation

From the column of cumulative frequency cf, we find that the 12th observation lies in the class 10-20.

∴ The median class is 10-20.

Now,
∴L=lower boundary point of median class =10

∴n=Total frequency =25

∴cf=Cumulative frequency of the class preceding the median class =4

∴f=Frequency of the median class =9

∴c=class length of median class =10

Median M=L+ (((n/2)-cf) / f ) * c

=10+ (12-4) / 9 *10

=10+(8 / 9) *10

=10+8.8889

=19.4444

(d) MEAN

Mean =A+ (∑fd / n) * h

=25 + (-9 / 25) *10

=25 + (-0.36) *10

=25 - 3.6

=21.4

(e) SAMPLE VARIANCE

Sample Variance s2= |- h2 n (-9)2 37- 25 102 24

=((37-3.24) / 24) * 100

= (33.76 / 24) * 100

=1.4067 * 100

=140.6667