To answer part (a), we would typically calculate the Pearson correlation coefficient (r) for the given data set to determine if there is a significant linear correlation between court incomes and justice salaries. If |r| is close to 1, it indicates a strong linear correlation, whereas a value close to 0 indicates a weak correlation.
For part (b), if there is a linear correlation, we would use the least squares method to find the line of best fit and then use that line to predict the salary for the justice in the town of Beekman, based on their court income of $83,941 (which should be converted to thousands of dollars as per the data given).
(a) To determine if there is a linear correlation between court incomes and justice salaries, we can calculate the correlation coefficient, also known as the Pearson's correlation coefficient. The correlation coefficient ranges from -1 to 1, where a value close to 1 indicates a strong positive linear correlation, a value close to -1 indicates a strong negative linear correlation, and a value close to 0 indicates no linear correlation.
To calculate the correlation coefficient, we need the court incomes and justice salaries data. Let x represent the court incomes and y represent the justice salaries. Using the given data, we can create a scatter plot and calculate the correlation coefficient:
(653, 40)
(415, 67)
(924, 41)
(131, 56)
(272, 46)
(252, 61)
(111, 51)
(543, 21)
(826, 25)
After calculating the correlation coefficient, we can determine if there is sufficient evidence to conclude a linear correlation.
(b) To find the best predicted salary for the justice in the town of Beekman based on the given court income of $83,941, we can use the regression line equation. This equation represents the line that best fits the data points and can be used to predict values based on the independent variable (court income).