3. Linear Regression Given the data set: \begin{tabular}{|c|c|c|c|c|} \hline $x$ & $y$ & $x^2$ & $y^2$ & $xy$ \\ \hline 1 & 2 & 1 & 4 & 2 \\ 2 & 3 & 4 & 9 & 6 \\ 3 & 5 & 9 & 25 & 15 \\ 4 & 7 & 16 & 49 & 28 \\ 5 & 8 & 25 & 64 & 40 \\ \hline 15 & 25 & 55 & 151 & 91 \\ \hline \end{tabular} (a) Calculate the correlation coefficient c. (b) Perform a simple linear regression to find the line $Y = b_1x + b_0$ of best fit. Plot the line. (c) Using the regression line obtained in part (a), predict the value of $Y_p$ when $X_p = 6$. Construct a 95% interval for this prediction.
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Step 1: Calculate the means of x and y: $\bar{x} = \frac{15}{5} = 3$ $\bar{y} = \frac{25}{5} = 5$ Show more…
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1) a) Linear Regression and Correlation. x y 1 2.51 2 -2.58 3 6.33 4 0.64 5 9.05 6 2.86 7 7.67 Compute the equation of the linear regression line in the form y = mx + b, where m is the slope and b is the intercept. Use at least 3 decimal places. (Round if necessary) y = __x + __ Compute the correlation coeficient for this data set. Use at least 3 decimal places. (Round if necessary) r= __ Compute the P-value (Use HA: slope ≠ 0 for the alternative hypothesis.) Use at least 3 decimal places. (Round if necessary) P-value = __ At the alpha = 0.05 significance level, is the correlation significant? Yes, significant correlation No b) Linear Regression and Correlation. x y 2 -3.26 3 7.96 4 0.08 5 -6.3 6 2.62 7 2.34 8 3.56 9 -8.62 Compute the equation of the linear regression line in the form y = mx + b, where m is the slope and b is the intercept. Use at least 3 decimal places. (Round if necessary) y = __x + __ Compute the correlation coeficient for this data set. Use at least 3 decimal places. (Round if necessary) r= ___ Compute the P-value (Use HA: slope ≠ 0 for the alternative hypothesis.) Use at least 3 decimal places. (Round if necessary) P-value = ___ At the alpha = 0.01 significance level, is the correlation significant? No Yes, significant correlation
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(a) Draw a scatter diagram of the data. What type of relation appears to exist between $x$ and $y ?$ (b) Find the quadratic regression equation $\hat{y}=b_{0}+b_{1} x+b_{2} x^{2}$ (c) Draw a residual plot against the fitted values, $x,$ and $x^{2}$. Also. draw a boxplot of the residuals. Are there any problems with the model? (d) Interpret the coefficient of determination. (e) Does the $F$ -test indicate that we should reject $H_{0}: \beta_{1}=\beta_{2}=0 ?$ Is either coefficient not significantly different from zero? (f) Construct and interpret $95 \%$ confidence and prediction intervals for $x=4$ $$ \begin{array}{cc} x & y \\ \hline 2.3 & 19.3 \\ \hline 2.7 & 14.8 \\ \hline 3.2 & 10.2 \\ \hline 4.1 & 4.8 \\ \hline 4.9 & 2.9 \\ \hline 5.6 & 3.9 \\ \hline 6.4 & 7.9 \\ \hline \end{array} $$
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