Question

From a survey of construction labor, the work duration (in number of hours) per day and the average productivity (percent efficiency) were recorded as follows:

Duration and Productivity (x, y) (hour, percent)	No. of Observations
6, 45	4
6, 65	3
6, 85	10
8, 45	5
8, 65	36
8, 85	24
10, 45	7
10, 65	23
10, 85	13
12, 45	8
12, 65	5
12, 85	2
Total	140

SHOW ALL WORK

A. Determine the marginal probability distribution function (PDF) for X, the distribution of work duration and plot graphically on p_x (x) versus x for y = 6, 8, 10 and 12. Present p_x (x) in terms of relative frequencies of the observations.

B. Similarly, determine the marginal PDF for Y, representing the distribution of productivity.

C. If the work duration per day is 10 hr, what is the probability that the average productivity will be 85%?

Answer 1

work duration (in number of hours) per day(X)	average productivity (percent efficiency) (Y)	No. of Observations
6	45	4
6	65	3
6	85	10
8	45	5
8	65	36
8	85	24
10	45	7
10	65	23
10	85	13
12	45	8
12	65	5
12	85	2
Total		140

A. Determine the marginal probability distribution function (PDF) for X, the distribution of work duration and plot graphically on px (x) versus x for y = 6, 8, 10 and 12. Present px (x) in terms of relative frequencies of the observations.

work duration (in number of hours) per day(X)	Frequency	Relative Frequency
6	4+3+10 = 17	17/140 = 0.1214
8	5+36+24 = 65	65/140 = 0.4643
10	7+23+13 = 43	43/140 = 0.3071
12	8+5+2 = 15	15/140 = 0.1072

b)

Average productivity (percent efficiency) (Y)	Frequency	Relative Frequency
45	4+5+7+8 = 24	24/140 = 0.1714
65	3+36+23+5 = 67	67/140 = 0.4786
85	10+24+13+2 = 49	49/140 = 0.35

C. If the work duration per day is 10 hr, what is the probability that the average productivity will be 85%?

P[ average productivity will be 85% | work duration per day is 10 hr ]= P[ average productivity will be 85% and work duration per day is 10 hr ] / P[ work duration per day is 10 hr ]

P[ average productivity will be 85% and work duration per day is 10 hr ] = 13/140

[ work duration per day is 10 hr ] = 43/140

P[ average productivity will be 85% | work duration per day is 10 hr ]= (13/140)/(43/140)

P[ average productivity will be 85% | work duration per day is 10 hr ]= 13/43

P[ average productivity will be 85% | work duration per day is 10 hr ]=0.3023