Question

In University of Harimau Malaya, Dr Ku is doing a research in developing a new drug for an allergy. He conducts an experiment to study how different dosages of the drug affect the duration of relief from the allergic symptoms. There are ten patients are included in the experiment. Each patient receives a specified dosage of the drug and asked to report back as soon as the protection of the drug seems to wear off. The observations are recorded in table below, which shows the dosage (in Milligrams) and duration of relief (in Days) for the 10 patients. Duration of Dosage, x Relief, y 4 4 5 6 7 10 6 13 10 15 17 23 19 25 23 8 9 9 10 i. Calculate the Ey, Ey. Ex: Zx and Xxx (1 mark) ii. Calculate correlation coefficient and interpret its value. (6 marks) iii. Calculate coefficient of determination and interpret its value. (2 marks) iv. Estimate the regression parameters and write the estimated linear regression model. (7 marks) Predict the duration of relief when the dosage 13 mg. (2 marks) vi. Predict the dosage used when the duration of relief is 12 days. (2 marks) V.

Answer 1

Solution:

Here

here from above

X= Dosage

Y=Duration of Relief

1)

	x	y	x2	y2	x*y
	4	10	16	100	40
	4	6	16	36	24
	5	13	25	169	65
	6	10	36	100	60
	7	15	49	225	105
	7	17	49	289	119
	8	23	64	529	184
	9	19	81	361	171
	9	25	81	625	225
	10	23	100	529	230
Total $\sum$	69	161	517	2963	1223

2) Correlation coefficient

	x	y	x-mean(x)	(x-mean(x))^2	y-mean(y)	(y-mean(y))^2	(x-mean(x))(y-mean(y))	y_pred	(y-pred(y)	(y-y_pred)^2
	4	10	-2.9	8.41	-6.1	37.21	17.69	13.516	-3.516	12.362256
	4	6	-2.9	8.41	-10.1	102.01	29.29	13.516	-7.516	56.490256
	5	13	-1.9	3.61	-3.1	9.61	5.89	13.682	-0.682	0.465124
	6	10	-0.9	0.81	-6.1	37.21	5.49	13.848	-3.848	14.807104
	7	15	0.1	0.01	-1.1	1.21	-0.11	14.014	0.986	0.972196
	7	17	0.1	0.01	0.9	0.81	0.09	14.014	2.986	8.916196
	8	23	1.1	1.21	6.9	47.61	7.59	14.18	8.82	77.7924
	9	19	2.1	4.41	2.9	8.41	6.09	14.346	4.654	21.659716
	9	25	2.1	4.41	8.9	79.21	18.69	14.346	10.654	113.507716
	10	23	3.1	9.61	6.9	47.61	21.39	14.512	8.488	72.046144
sum	69	161	-3.55271E-15	40.9	-1.4E-14	370.9	112.1	139.974	21.026	379.019108
mean	6.9	16.1

$R=r=\frac{\sum (x-\overline{x})(y-\overline{y})}{\sqrt{(x-\overline{x})^{2}(y-\overline{y})^{2}}}$

112.1 R= 40.9 * 370.9

R = 0.910155

Correaltion coefficient is 0.910 is stated that the dosage and duration of relief is positively correlated

3) Coefficient of Determination:

Coefficient of Determination = R^2 = 0.910155^2 = 0.8283

coefficient of determination of 82.83% shows that 82.83% of the data fit the regression model

4) linear regression line

y = x+b =

hence the parameter are

bo = y-intercept of the line

b1= slope of the line

bo=y - bix

b, Σ(α - T)(y - g) Σε – 7)2

b1= 112.1 / 40.9 = 2.740

b1= slope of the line is 2.740

b0 = 16.1 -  2.740 * 6.9 = -2.806

b0 = y-intercept of the line is -2.806

therefore,linear regression model is

$\widehat{y}=b_{1}X+b_{0}$

$\widehat{y}=2.740X-2.806$

5) predict the duration of relief when dosage is 13 mg

$\widehat{y}=2.740*13-2.806$

$\widehat{y}= 32.814$

Predicted the duration is 15.36 ~ 15 days

6)  predict the dosage used when the durattion of relief is 12 days

X = 5.4036

predicted dosage is 5..40 mg