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1. Regression and Correlation An experiment was conducted to study the effect of increasing the dosage of a on sleeping time. Three readings were made at each of three dose levels certain barbiturate Sleeping Time (Hours) Dosage (11 μM/kg) 4 10 10 9 15 15 15 13 9 ΣΥ-642, ΣXY = 780 2X 84, ΣΧ2 1002, ΣΧΥ 780 ΣΥ-72, a. Plot the scatter Diagram b. Determine the Correlation Coefficient r and the coefficient of determination R2n c. Determine the equation of the regression line relating dosage (X) to sleeping time (Y) d. Place a 95 percent confidence interval on β1 e. Test the hypothesis of no linear relationship between variables. (Use α-- 0.05) f. What is the predicted sleeping time for a dose of 12 μM/kg?

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Answer #1

(a) Scatter plot

4 6 4 0 42086420 au 11 Buldaals

(b) Correlation coefficient

The formula of the coefficient of correlation is,

r = rac{nsum XY - sum Xsum Y}{sqrt{[nsum X^{2}-(sum X)^{2}][nsum Y^{2}-(sum Y)^{2}]}}

Where n = 9

By plugging the values of all sums,

9 780-84 72 9 1002- (84)219 642 - (72)2

A coefficient of determination R2

R2 = r * r = 0.8107

(c) The equation of the regression line:

The general equation of the regression line is,

y=30 + 314.

Where y - dependent variables which is sleeping time

x - an independent variable which is dosage.

Bo-y intercept

31-slope

The formula to find the slope and intercept is,

nXx2ー(ΣΧ)2 9*1002ー(84)2 = 0.4954

Σ yーβι Σ x 72-0.4954 * 84 Во 3.3761

The regression equations become,

y = 3.3761 + 0.4954x

(d) 95% confidence interval for beta1 that is slope

The formula of confidence interval for slope is,

eta _{1}pm t_{n-2}S_{eta _{1}}

Se SBS Svn-

125-113 EN -У,)2 = 1.3361 Se- 2-2

S_{x} =sqrt{rac{sum (X - ar X)^2}{n-1}} = 5.2202

t is the critical value at alpha 0.05 with degrees of freedom = n - 2 = 7

t = 2.365

Plugging all the values in the formula of the standard error of slope,

S_{eta _{1}} = rac{S_{e}}{S_xsqrt{n-1}} = 0.09049

The confidence interval is,

eta _{1}pm t_{n-2}S_{eta _{1}}= 0.4954 pm 2.365*0.09049

(0.2814, 0.7094), is the 95% confidence interval for slope.

(e) Test of linear relationship

The null and alternative hypothesis are:

H0: There is no linear relationship between the variables.

H1: There is a linear relationship between the variables.

Test statistics formula

t = rsqrt{rac{n-2}{1-r^{2}}} = 5.475

t critical value is found above as 2.365

Here test statistics > critical value, so reject the null hypothesis.

That is, there is a linear relationship between the variables.

(f) Predicted value at x = 12

y = 3.3761 + 0.4954x = 3.3761 + 0.4954 * 12 = 9.3209

Predcited sleeping time is approximately 9 hours.

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