Answer to Question #161821 in Statistics and Probability for Lemchi Divine

Question #161821

The table below shows the average body weight 'W' and food consumption 'F'


W- 5.1 4.6 5.1. 4.8 4.4 5.9 4.7 5.1

F- 87.1|93.1|89.8|91.4|95.5|92.1|95.5|99.3

W- 5.2 4.9

F- 93.4|94.4


1. Plot the scatter diagram for the data

2. Fit the least squares regression equation of W on F.

3. Using the fitted regression equation, estimate the average body weights 'W' given that food 'F' is 6.0

4. Compute the Spearman Rank Correlation Coefficient and comment on your results.


1
Expert's answer
2021-02-24T06:40:46-0500

1.




2.

xˉ=931.610=93.16,yˉ=49.810=4.98\bar{x}=\dfrac{931.6}{10}=93.16, \bar{y}=\dfrac{49.8}{10}=4.98


SSxx=xi2nxˉ2=102.484SS_{xx}=\sum x_i^2-n\bar{x}^2=102.484

SSyy=yi2nyˉ2=1.536SS_{yy}=\sum y_i^2-n\bar{y}^2=1.536

SSxy=xiyinxˉyˉ=3.088SS_{xy}=\sum x_iy_i-n\bar{x}\bar{y}=-3.088

b=SSxySSxx=3.088102.484=0.03013b=\dfrac{SS_{xy}}{SS_{xx}}=\dfrac{-3.088}{102.484}=-0.03013

a=yˉbxˉ=4.98(0.03013)(93.16)=7.78705a=\bar{y}-b\bar{x}=4.98-(-0.03013)(93.16)=7.78705

W=a+bFW=a+bF

W=7.787050.03013FW=7.78705-0.03013F

3. Given F=6F=6


W(6)=7.787050.03013(6)=7.6W(6)=7.78705-0.03013(6)=7.6

4. Rank the two data sets. Ranking is achieved by giving the ranking '1' to the biggest number in a column, '2' to the second biggest value and so on. The smallest value in the column will get the lowest ranking. This should be done for both sets of measurements.


Rank(x)Rank(y)175227348.514108.531076975\begin{matrix} Rank(x) & Rank(y)\\ 1 & 7\\ 5 & 2 \\ 2 &7 \\ 3 & 4 \\ 8.5 & 1 \\ 4 & 10 \\ 8.5 & 3 \\ 10 & 7 \\ 6 & 9 \\ 7 & 5 \end{matrix}

d2=36+9+25+1+56.25+36+30.25\sum d^2=36+9+25+1+56.25+36+30.25

+9+9+4=215.5+9+9+4=215.5

rs=16d2n3n=16(215.5)10310=0.3061r_s=1-\dfrac{6\sum d^2}{n^3-n}=1-\dfrac{6(215.5)}{10^3-10}=-0.3061

Or

SSxˉxˉ=Rank(xi)21n(Rank(xi))2SS_{\bar{x}\bar{x}}=\sum Rank(x_i)^2-\dfrac{1}{n}(\sum Rank(x_i))^2

=384.5110(55)2=82=384.5-\dfrac{1}{10}(55)^2=82

SSyˉyˉ=Rank(yi)21n(Rank(yi))2SS_{\bar{y}\bar{y}}=\sum Rank(y_i)^2-\dfrac{1}{n}(\sum Rank(y_i))^2

=383110(55)2=80.5=383-\dfrac{1}{10}(55)^2=80.5

SSxˉyˉ=Rank(xi)Rank(yi)SS_{\bar{x}\bar{y}}=\sum Rank(x_i) Rank(y_i)

1n(Rank(xi))(Rank(yi))-\dfrac{1}{n}(\sum Rank(x_i))(\sum Rank(y_i))

=276110(55)2=26.5=276-\dfrac{1}{10}(55)^2=-26.5


rs=SSxˉyˉSSxˉxˉSyˉyˉ=26.58280.5=0.3262r_s=\dfrac{SS_{\bar{x}\bar{y}}}{\sqrt{SS_{\bar{x}\bar{x}}}\sqrt{S_{\bar{y}\bar{y}}}}=\dfrac{-26.5}{\sqrt{82}\sqrt{80.5}}=-0.3262

Negative correlation. The correlation is too weak to be thought significant.



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