STA301 2nd Assignment solution spring 2012

Question 1:                                                                                                                           Marks: 7
For the given cumulative frequency table of students of different age groups, calculate the coefficient of standard deviation and coefficient of variation.

Age in years	Cumulative frequency of students (cf)
5-8	3
9-12	15
13-16	24
17-20	51
21-24	57
25-28	60

Solution:

Age in Years	C.F	X	F	Fx	Fx2
5 – 8	3	6.5	3	19.5	126.5
9 – 12	15	10.5	12	126	1323
13 – 16	24	14.5	9	130.5	1892.25
17 – 20	51	18.5	27	499.5	9240.75
21 – 24	57	22.5	6	135	3037.5
25 – 28	60	26.5	3	79.5	2106.75
Total	210	99	60	990	17726.75

Variance =S2 = ∑fx2 / ∑f – (∑fx/∑f)2
17726.75 / 60 – (990/60)2
295.44583 – (16.5)2
295.44583 – 272.25
23.19583
Standard Deviation = √S2 = √23.19583 = 4.816204938

Coefficient of Standard Deviation = S/Mean
_
Mean= X = 990/60 = 16.5
S = 4.81620
So                                            4.816204938 / 16.5 = 0.291891208
Coefficient of Variation = 4.81620 / 16.5
= 0.291891208 * 100
Coefficient of Variation = 29.1891208

Question 2:                                                                                                                           Marks: 8
From the following data of hours worked in a factory (x) and output units (y), determine the regression line of y on x, the linear correlation coefficient and interpret the result of correlation coefficient.

Hours (X)	91	102	83	93	89	72	82	85	79
Production (Y)	300	302	315	330	300	250	300	340	315

Solution:

X	Y	X.Y	X2	Y2
91	300	27300	8281	90000
102	302	30804	10404	91204
83	315	26145	6889	99225
93	330	30690	8649	108900
89	300	26700	7921	90000
72	250	18000	5184	62500
82	300	24600	6724	90000
85	340	28900	7225	115600
79	315	24885	6241	99225
776	2752	238024	67518	846654

∑X =     776
∑Y =     2752
∑X.Y = 238024
∑X2   = 67518
∑Y2   = 846654

Regression line Y on X
Byx  =   n∑xy – (∑x) (∑y) / n∑ x2 – (∑x)2
= 9(238024) – (776)(2752) / 9 (67518) – (776)2
= 9(238024) – (2135552) / 9 (67518) – (602176)
= 2142216 – 2135552 / 607662 – 602176
= 6664 / 5486
= 1.2147284
_      _
A  = y –  bx
= ∑y / n – b (∑x/n)
= 2752 /9 – 1.2147284 (776/9)
= 305.77777 – 1.2147284 (86.22222)
= 201.0411907
_
Y= a+bx
= 201.0411904 + 1.2147284

Linear Coefficient of Correlation:

∑xy – (∑x)(∑y)/n
r = ——————————————–
√ [∑x2 – (∑x)2 / n ] [∑y2 – (∑y)2 / n]

(238024) – (776)(2752) /9
=  —————————————————–
√ [67518 – (776)2 / 9] [846654 – (2752)2 / 9]

238024 – 237283.5556
= ——————————————————-
√ (67518 – 66908.4444) (846654 – 841500.4444)
740.4444
= ————————–
√ (609.5556) (5153.556)
740.4444
= ——————-
√3141378.92
740.444
= ——————
1772.393557
= 0.417765003