A researcher wants to see whether there is a relationship between the number of

Question

A researcher wants to see whether there is a relationship between the number of absences and the final grades (on a scale of 0 to 10) earned by students in a statistics course. She selected ten students and collected data on the number of absences of these ten students (X rightarrow Independent variable) and their final grades (Y rightarrow Dependent variable). The following are the summary of the data: sigma^n _i = 1 x_i = 84: sigma^n _i = 1 y_i = 72: sigma^n _i = 1 x_i y_i = 549: sigma^n _i = 1 x^2 _i = 912: sigma^n _i = 1 y^2 _i = 549. (a) Find the regression line y = b_0 + b_1 x, that is, find b_0 and b_1. (b) What percentage of the variation in Y is explained by X ? (c) Conduct the appropriate statistical test of hypotheses to check whether the linear relationship between Y and X is significant. Use alpha = 0.05.

Explanation / Answer

here number of observation=n=10

Sxx=sum(x2)-(sum(x))2/n=912-84*84/10=206.4

Syy=sum(y2)-(sum(y))2/n=549-72*72/10=30.6

Sxy=sum(xy)-sum(x)*sum(y)/n=549-84*72/10=-55.8

(a) b1=Sxy/Sxx=-55.8/206.4=-0.27

b0=sum(y)/n - b1*sum(x)/n=72/10 -(-0.27)*84/10=7.2 -(-0.27)*8.4=9.468

y=9.468-0.27x

(b) correlation coefficient between x and y=r=Sxy/sqrt(Sxx*Syy)=-55.8/sqrt(206.4*30.6)=-0.7021

proportion of variation explained by X=r*r=(-0.7021)*(-0.7021)=0.4929

required % of variation in Y is explained by X=49.29

(c) here we use t-test wheter the r is significant and

t=r*sqrt(n-2)/sqrt(1-r*r)=-0.7021*sqrt(10-2)/sqrt(1-0.4929)=-2.79 with n-2=10-2=8 df

critical t(0.05,8)=2.3 is less than absolute value of t=2.79 so we conclude there is significant relationship between X and Y at alpha=0.05

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