Question

ch 5 in a study, nine tires of a particular brand were driven on a track under identical conditions. each tire was driven a particular controlled distance (measured in thousands of miles) and the tread depth was measured after the drive. tread depth is measured in \mils.\ here, 1 mil is 0.001 inch. the least - squares regression line was computed, and added to a scatterplot of these data. on the plot, one data point is marked with an \x.\the equation of the least - squares regression line is:tread depth = 360.64 - 11.39x (thousands of miles)also, r² = 0.953. which of the following statements is true?according to the least - squares regression line, we would predict a decrease in groove depth of 11.39 mils for each 1000 miles driven on a tire.according to the least - squares regression line, the groove depth of a new tire (driven 0 miles) is predicted to be 360.64 mils.about 95.3% of the variation in tread depth is explained by the regression on miles.

Explanation:

Step1: Analyze the first option

The regression line is \( \text{Tread depth} = 360.64-11.39x\) (where \(x\) is in thousands of miles). The slope is \(- 11.39\). The slope of a regression line \(y = a+bx\) (in our case \(y=\text{Tread depth}\), \(a = 360.64\), \(b=-11.39\), \(x\) is miles in thousands) represents the change in \(y\) for a one - unit change in \(x\). So for each additional \(1000\) miles (\(x\) increases by \(1\)), the tread depth (in mils) decreases by \(11.39\).

Step2: Analyze the second option

When \(x = 0\) (new tire, \(0\) miles driven), substitute \(x = 0\) into the regression equation \(y=360.64-11.39x\). We get \(y=360.64-11.39\times0=360.64\) mils.

Step3: Analyze the third option

The coefficient of determination \(r^{2}\) measures the proportion of the variance in the dependent variable (tread depth) that is predictable from the independent variable (miles). Given \(r^{2}=0.953 = 95.3\%\), it means that about \(95.3\%\) of the variation in tread depth is explained by the regression on miles.

Answer:

All three statements are True.

1. According to the least - squares regression line, we would predict a decrease in groove depth of \(11.39\) mils for each \(1000\) miles driven on a tire.
2. According to the least - squares regression line, the groove depth of a new tire (driven \(0\) miles) is predicted to be \(360.64\) mils.
3. About \(95.3\%\) of the variation in tread depth is explained by the regression on miles.