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Consider the data given in the following table. $$ \begin{array}{l|llllll} \hline x & 10 & 20 & 30 & 40 & 50 & 60 \\ \hline y & 12 & 15 & 19 & 21 & 25 & 30 \\ \hline \end{array} $$ a. Find the least squares regression line and the linear correlation coefficient \(r\). b. Suppose that each value of \(y\) given in the table is increased by 5 and the \(x\) values remain unchanged. Would you expect \(r\) to increase, decrease, or remain the same? How do you expect the least squares regression line to change? c. Increase each value of \(y\) given in the table by 5 and find the new least squares regression line and the correlation coefficient \(r\). Do these results agree with your expectation in part b?

Step by step solution

01

Calculate the Least Squares Regression Line and Correlation Coefficient

First, we calculate the sums necessary for the calculations: \( \Sigma x\), \( \Sigma y\), \( \Sigma x^2\), \( \Sigma y^2\), \( \Sigma xy\). \nNext, with n as the number of pairs (here, n=6), the formulas for the slope(b) and the y-intercept(a) of the regression line are : \nb = \((n\Sigma xy - \Sigma x \Sigma y) / (n\Sigma x^2 - (\Sigma x)^2)\) \na = \((\Sigma y - b\Sigma x) / n\) \nThe correlation coefficient r can be calculated as follows: \nr = \((n\Sigma xy - \Sigma x \Sigma y) / \sqrt{((n\Sigma x^2 - (\Sigma x)^2)(n\Sigma y^2 - (\Sigma y)^2))}\)

02

Predict the Changes to the Regression Line and Correlation Coefficient

Adding the same constant value to each data point won't affect the slope of the regression line or the correlation coefficient, since these are measures of the relationship between x and y, which won't change. The y-intercept will increase by 5 because it represents the predicted value of y when x=0, and we're adding 5 to all y-values.

03

Recalculate the Regression Line and Correlation Coefficient

Increase each y value in the table by 5 and repeat the calculations in Step 1.

04

Compare Results

Compare the new values of the correlation coefficient and the regression line from step 3 to the initial predictions made in step 2 to see if they align.

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