Linear Statistical Models and Hypothesis Testing
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1.
a) Find the sample correlation coefficient r between the league positions in 2008 and 2009. Test the null hypothesis that the population correlation coefficient p between the league positions in 2008 and 2009 is zero.
b) Obtain an approximate 95% confidence interval for the population correlation coefficient between the league positions in 2008 and 2009. Comment.
c) State the assumptions of the simple linear regression model.
d) Fit a simple linear regression model to the data in Table 1.1 to explain the league position in 2009 as linear function of the league position in 2008.
e) Calculate the percentage of variation of y explained by x.
f) Obtain the fitted value, residual and standardized residual for Leicester. Comment.
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The linear statistical models and hypothesis testings is examined in the solution.
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a) Mean of x: (7+1+10+4...+2)/11=6
SSx=(7-6)^2+(1-6)^2+...+(2-6)^2=110
Mean of y: (1+2+3+...+11)/11=6
SSy=(1-6)^2+(2-6)^2+...+(11-6)^2=110
Sxy=(1-6)*(7-6)+(2-6)*(1-6)+...+(2-6)*(11-6)=-7
r=Sxy/sqrt(SSx*SSy)=-7/sqrt(110*110)=-0.06364
Null hypothesis: r=0
Alternative hypothesis: r<0
This is one tailed t test.
The degree of freedom is n-2=11-2=9.
The critical t value is -1.83
Test value t=r*sqrt((n-2)/(1-r^2))=-0.06364*sqrt((11-2)/(1-0.06364^2))=-0.19
Since -0.19>-1.83, we could not reject null hypothesis.
Based on the test, we could conclude that population correlation coefficient between the league ...
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