Question

in a statistics class, a teacher had the students complete an activity in which they grabbed as many bite - sized pretzels as they could with their dominant hand, without crushing them. the teacher then measured their handspan in centimeters. the scatterplot displays the data the teacher collected along with the least - squares regression line. one student with a handspan of 23 cm grabbed 38 pretzels. this point is circled on the graph.
what effect will the circled point have on the slope of the least - squares regression line?
it will increase the value of the slope because its residual is a large positive value.
it will decrease the value of the slope because its residual is a large positive value.
it will increase the value of the slope because its residual is a large negative value.
it will decrease the value of the slope because its residual is a large negative value.

Explanation:

A residual is calculated as \( \text{Residual}=y - \hat{y}\) (where \(y\) is the observed value and \(\hat{y}\) is the predicted value). If the point is above the regression line, \(y>\hat{y}\) (positive residual). The regression line tries to minimize the sum of squared residuals. A point with a large positive residual pulls the line upwards on the right - hand side (since the \(x\) - value (handspan) is relatively large). When we calculate the slope of the regression line \(b_1=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})(y_i - \bar{y})}{\sum_{i=1}^{n}(x_i-\bar{x})^2}\), a large positive residual for a relatively large \(x\) - value (\(x = 23\) is larger than many other \(x\) - values in the scatter - plot) increases the numerator \(\sum_{i = 1}^{n}(x_i-\bar{x})(y_i - \bar{y})\) (because \((x_i-\bar{x})\) is positive for \(x_i\) above the mean \(x\) and \(y_i-\hat{y}_i\) (and approximately \(y_i - \bar{y}\) for non - extreme cases) is positive). An increase in the numerator of the slope formula \(b_1\) (while the denominator \(\sum_{i = 1}^{n}(x_i-\bar{x})^2\) is less affected) leads to an increase in the slope.

Answer:

It will increase the value of the slope because its residual is a large positive value.