1) Let's start by making a table for that:
So now let' calculate the mean X bar , and Y bar:
[tex]\begin{gathered} x=\frac{88+75+76+92+96+94+83+90+99+65+77+88+82+83+94+97}{16} \\ \bar{x}=86.1875 \end{gathered}[/tex]Similarly for the average of y:
[tex]\bar{y}=7.625[/tex]And now let's calculate the Standard Deviation for that sample Sx, and Sy:
[tex]\begin{gathered} S_x=\sqrt[]{\frac{\sum^{}_{}(x_i-\bar{x})^2}{n-1}}=9.502411975 \\ S_y=\sqrt[]{\frac{\sum^{}_{}(y_i-\bar{y})^2}{n-1}}=1.190238071 \end{gathered}[/tex]And finally, let's calculate the correlation coefficient: One over n-1 times the Summation of the Standard deviations of the sample:
[tex]\begin{gathered} r=\frac{1}{n-1}\cdot\sum ^{}_{}(\frac{x_i-\bar{x}}{S_x})(\frac{y-\bar{y}}{S_y}) \\ r=0.8318499656 \end{gathered}[/tex]a) We can now start to write out the Least Squares Regression equation is as it follows calculating the slope
[tex]b_1=r\cdot\frac{S_y}{S_x}\Rightarrow b_1=0.8318499656\cdot\frac{1.190238071}{9.502411975}=0.104[/tex]And the linear coefficient (y-intercept)
[tex]\begin{gathered} b_0=\bar{y}-b_1\bar{x} \\ b_0=7.625-0.104\cdot86.1875 \\ b_0=-1.3385 \\ \hat{y}=-1.3385+0.104x \end{gathered}[/tex]b) Since the number of hours of sleeping on average is given we can plug into y the number of hours to get x the score, Just like that:
[tex]\begin{gathered} \hat{y}=b_0+b_1x \\ \hat{y}=-1.3385+0.104x \\ 8=-1.3385+0.104x \\ 8+1.3385=0.104x \\ \frac{9.3385}{0.104}=\frac{0.104x}{0.104} \\ x\approx89.79 \end{gathered}[/tex]3) Hence the answer is
[tex]\begin{gathered} a)\text{ }\hat{y}=-1.3385+0.104x \\ b)\text{ 89.79} \end{gathered}[/tex]
