Question

1. A financial manager speculates about the relationship between family incomes and their allocation for investment. The following table presents the results of a survey of 8 randomly selected families.

Annual income ‘000’ ksh(X)	8	12	9	24	13	37	10	16
Per cent allocation for investment(Y)	36	25	33	15	28	19	20	22

1. Construct a scatter diagram for the data and comment on the relationship      between income and allocation for investment
2. Develop a regression equation that best describes this data
3. Estimate the percent allocated for investment if the annual income of a family is Ksh 25,000
4. Compute coefficient of determination and interpret the results.
5. Compute the Karl Pearson product-moment correlation co-efficient and interpret the results.
6. Compute the spearman rank correlation co-efficient and interpret the results
7. VII. Compare the two measures of relationship i.e. Spearman and Pearson correlation co-efficient

Answer 1

1. Construct a scatter diagram for the data and comment on the relationship      between income and allocation for investment

2.Develop a regression equation that best describes this data

	Income	Percent	Income*Percent	Income2	Percent2
	8	36	288	64	1296
	12	25	300	144	625
	9	33	297	81	1089
	24	15	360	576	225
	13	28	364	169	784
	37	19	703	1369	361
	10	20	200	100	400
	16	22	352	256	484
Sum =	129	198	2864	2759	5264

Based on the above table, the following is calculated:

Therefore, based on the above calculations, the regression coefficients (the slope m, and the y-intercept n) are obtained as follows:

Therefore, we find that the regression equation is:

Percent = 32.5586 - 0.4843 Income

Y=32.5586-0.4843 X

3.Estimate the percent allocated for investment if the annual income of a family is Ksh 25,000

X=25

Y=32.5586-0.4843 X

Y=32.5586-0.4843*25

Y=20.4511 percent.

4.Compute coefficient of determination and interpret the results.

R²=

R²=0.4382

hence, the variability expalined by the model is 0.4382 or 43.82%

5.Compute the Karl Pearson product-moment correlation co-efficient and interpret the results.

Therefore, based on this information, the sample correlation coefficient is computed as follows

r=−0.662

there is a negative and moderatly weak relation between income and percent.

6.Compute the spearman rank correlation co-efficient and interpret the results

	Rank(X)	Rank(Y)	Rank(X)*Rank(Y)	Rank(X)2	Rank(Y)2
	1	8	8	1	64
	4	5	20	16	25
	2	7	14	4	49
	7	1	7	49	1
	5	6	30	25	36
	8	2	16	64	4
	3	3	9	9	9
	6	4	24	36	16
Sum =	36	36	128	204	204

The Spearman correlation coefficient rS​ is computed using the same calculations as those used for the computation of Pearson's correlation, but in this case, the ranks are used instead of the actual values. The following formula is used:

rS​=SSXX​SSYY​​SSXY​​

where

Therefore, based on this information, the sample Spearman correlation coefficient is computed as follows

rs=−0.81

strong negative relationship between income and percent.

please rate my answer and comment for doubts.