Question

given is a scatterplot of the number of problems people have as it relates to their net worth (in thousands of dollars). the dot in the bottom - right corner represents a person with a net worth of $900,000 that has relatively few of problems. the point in the bottom right corner is an outlier. describe the effect this point has on the equation of the least - squares regression line. this point decreases the slope and decreases the y - intercept of the least - squares regression line. this point increases the slope and increases the y - intercept of the least - squares regression line. this point decreases the slope and increases the y - intercept of the least - squares regression line. this point increases the slope and decreases the y - intercept of the least - squares regression line. option a option b option c option d

Explanation:

Step1: Understand the relationship between net - worth and number of problems

The general trend (without the outlier) shows that as net - worth (x - value) increases, the number of problems (y - value) increases. The outlier has a large x - value (\(x = 900\)) and a small y - value.

Step2: Analyze the effect on slope

The slope of the regression line \(b=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}\). The outlier has \(x_{i}\) (net - worth) greater than \(\bar{x}\) and \(y_{i}\) (number of problems) less than \(\bar{y}\). So \((x_{i}-\bar{x})(y_{i}-\bar{y})\) is negative for the outlier. This reduces the numerator of the slope formula, thus decreasing the slope.

Step3: Analyze the effect on y - intercept

The y - intercept \(a=\bar{y}-b\bar{x}\). Since \(b\) (slope) is decreased and \(\bar{x}\) and \(\bar{y}\) are affected by the outlier. Let's consider the formula \(y = a+bx\). If we have a point \((x_{0},y_{0})\) (the outlier with large \(x_{0}\) and small \(y_{0}\)), substituting into \(y=a + bx\) gives \(y_{0}=a+bx_{0}\). Solving for \(a\): \(a=y_{0}-bx_{0}\). Since \(b\) is made smaller (from step 2) and \(x_{0}\) is large and \(y_{0}\) is small, the value of \(a\) (y - intercept) is increased.

Answer:

This point decreases the slope and increases the \(y\) - intercept of the least - squares regression line. So the answer is Option C.