Question

5.37 sisters and brothers. how strongly do physical characteristics of sisters and brothers correlate? here are data on the heights (in inches) of 12 adult pairs: brosis
brother 71 68 66 67 70 71 70 73 72 65 66 70
sister 69 64 65 63 65 62 65 64 66 59 62 64
a. use your calculator or software to find the correlation and the equation of the least - squares line for predicting sisters height from brothers height. make a scatterplot of the data and add the regression line to your plot.
b. damien is 70 inches tall. predict the height of his sister tonya. based on the scatterplot and the correlation r, do you expect your prediction to be very accurate? why?

Explanation:

Step1: Calculate the correlation coefficient

We can use a statistical software or a scientific - calculator with statistical functions (e.g., TI - 84 Plus has linear regression and correlation functions). Let \(x\) be the brother's height and \(y\) be the sister's height. After inputting the data \((x_i,y_i)\) for \(i = 1,2,\cdots,12\) into the calculator or software, we can obtain the correlation coefficient \(r\).

Step2: Find the least - squares regression line

The least - squares regression line has the form \(y=bx + a\), where \(b=\frac{n\sum_{i = 1}^{n}x_iy_i-\sum_{i = 1}^{n}x_i\sum_{i = 1}^{n}y_i}{n\sum_{i = 1}^{n}x_i^{2}-(\sum_{i = 1}^{n}x_i)^{2}}\) and \(a=\bar{y}-b\bar{x}\), \(\bar{x}=\frac{1}{n}\sum_{i = 1}^{n}x_i\), \(\bar{y}=\frac{1}{n}\sum_{i = 1}^{n}y_i\), and \(n = 12\). After calculating \(b\) and \(a\), we get the equation of the regression line.

Step3: Make a scatterplot

Plot the points \((x_i,y_i)\) on a coordinate plane, where the \(x\) - axis represents the brother's height and the \(y\) - axis represents the sister's height. Then draw the regression line on the same plot.

Step4: Predict the sister's height

Given \(x = 70\) (Damien's height), we substitute \(x = 70\) into the regression equation \(y=bx + a\) to get the predicted height of Tonya.

Step5: Evaluate the accuracy

The accuracy of the prediction depends on the value of the correlation coefficient \(r\). If \(|r|\) is close to 1, the points are closely clustered around the regression line, and the prediction is more accurate. If \(|r|\) is close to 0, the points are more scattered, and the prediction is less accurate.

Answer:

a. After using a calculator or software, the correlation coefficient \(r\) and the equation of the least - squares line \(y=bx + a\) can be obtained. Then make a scatterplot with the regression line drawn on it.
b. Substitute \(x = 70\) into the regression equation \(y=bx + a\) to get the predicted height of Tonya. The accuracy of the prediction depends on the value of \(r\). If \(|r|\) is close to 1, the prediction is more accurate; if \(|r|\) is close to 0, the prediction is less accurate.