Listed below are systolic blood pressure measurements (in $\mathrm{mm} \mathrm{Hg}$ ) obtained from the same woman. Find the regression equation, letting the right arm blood press be the predictor $(\mathrm{x}$ ) variable. Find the best predicted systolic blood pressure in the left arm given that the systolic blood pressure in the right arm is $90 \mathrm{~mm} \mathrm{Hg}$. Use significance level of 0.05 .
\begin{tabular}{l|rrrrr}
Right Arm & 101 & 100 & 92 & 80 & 79 \\
\hline Left Arm & 176 & 171 & 144 & 146 & 145
\end{tabular}
Click the icon to view the critical values of the Pearson correlation coefficient $r$

Step 1 ：Given the systolic blood pressure measurements in the right and left arms, we are to find the regression equation and use it to predict the systolic blood pressure in the left arm given that in the right arm is \(90 \mathrm{~mm} \mathrm{Hg}\).

Step 2 ：The regression equation is of the form \(y = mx + c\), where \(m\) is the slope of the line and \(c\) is the y-intercept.

Step 3 ：The slope \(m\) can be calculated as \(m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}\) and the y-intercept \(c\) can be calculated as \(c = \frac{\sum y - m(\sum x)}{n}\), where \(x\) and \(y\) are the values of the predictor and response variables respectively, and \(n\) is the number of observations.

Step 4 ：Given the values of \(x = [101, 100, 92, 80, 79]\) and \(y = [176, 171, 144, 146, 145]\), and \(n = 5\), we calculate the slope \(m = 1.2717879604672058\) and the y-intercept \(c = 41.4303683737646\).

Step 5 ：Thus, the regression equation is \(y = 1.27x + 41.43\).

Step 6 ：We can use this regression equation to predict the systolic blood pressure in the left arm given that the systolic blood pressure in the right arm is \(90 \mathrm{~mm} \mathrm{Hg}\).

Step 7 ：Substituting \(x = 90\) into the regression equation, we get the predicted systolic blood pressure in the left arm as \(y = 155.89 \mathrm{~mm} \mathrm{Hg}\).

Step 8 ：Final Answer: The best predicted systolic blood pressure in the left arm given that the systolic blood pressure in the right arm is \(90 \mathrm{~mm} \mathrm{Hg}\) is \(\boxed{155.89}\).