Question

name u2 l3 d3—using a linear regression equation date the annual profits for a company are given in the following table, where x represents the number of years since 2002, and y represents the profit in thousands of dollars. write the linear regression equation that represents this set of data, rounding all coefficients to the nearest tenth. using this equation, estimate the calendar year in which the profits would reach 264 thousand dollars. years since 2002 (x) profits (y) (in thousands of dollars) 0 137 1 150 2 152 3 156 linear regression equation: final answer: the number of newly reported crime cases in a county in new york state is shown in the accompanying table, where x represents the number of years since 2010, and y represents number of new cases. write the linear regression equation that represents this set of data, rounding all coefficients to the nearest tenth. using this equation, find the projected number of new cases for 2021, rounded to the nearest whole number. years since 2010 (x) new cases (y) 0 1151 1 1131 2 1198 3 1190 4 1251 5 1236 linear regression equation: final answer: s-id.6a—summarize, represent, and interpret data on two categorical and quantitative variables. represent data on two quantitative variables on a scatter plot and describe how the variables are related. fit a least squares regression line to linear data using technology. use the fitted function to solve problems.

Explanation:

Step 1: Calculate the mean of x and y for the profit data

For the profit data:
\( x \) values: \( 0, 1, 2, 3 \)
\( \bar{x} = \frac{0 + 1 + 2 + 3}{4} = \frac{6}{4} = 1.5 \)
\( y \) values: \( 137, 150, 152, 156 \)
\( \bar{y} = \frac{137 + 150 + 152 + 156}{4} = \frac{595}{4} = 148.75 \)

Step 2: Calculate the slope (m) for the profit data

\( m = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2} \)
\( (x_0 - \bar{x})(y_0 - \bar{y}) = (0 - 1.5)(137 - 148.75) = (-1.5)(-11.75) = 17.625 \)
\( (x_1 - \bar{x})(y_1 - \bar{y}) = (1 - 1.5)(150 - 148.75) = (-0.5)(1.25) = -0.625 \)
\( (x_2 - \bar{x})(y_2 - \bar{y}) = (2 - 1.5)(152 - 148.75) = (0.5)(3.25) = 1.625 \)
\( (x_3 - \bar{x})(y_3 - \bar{y}) = (3 - 1.5)(156 - 148.75) = (1.5)(7.25) = 10.875 \)
\( \sum (x_i - \bar{x})(y_i - \bar{y}) = 17.625 - 0.625 + 1.625 + 10.875 = 29.5 \)
\( (x_0 - \bar{x})^2 = (0 - 1.5)^2 = 2.25 \)
\( (x_1 - \bar{x})^2 = (1 - 1.5)^2 = 0.25 \)
\( (x_2 - \bar{x})^2 = (2 - 1.5)^2 = 0.25 \)
\( (x_3 - \bar{x})^2 = (3 - 1.5)^2 = 2.25 \)
\( \sum (x_i - \bar{x})^2 = 2.25 + 0.25 + 0.25 + 2.25 = 5 \)
\( m = \frac{29.5}{5} = 5.9 \)

Step 3: Calculate the y-intercept (b) for the profit data

\( b = \bar{y} - m\bar{x} = 148.75 - 5.9(1.5) = 148.75 - 8.85 = 139.9 \) (rounded to the nearest tenth)
So the linear regression equation for profit is \( y = 5.9x + 139.9 \)

Step 4: Find the year when profit reaches 264 thousand dollars

Set \( y = 264 \)
\( 264 = 5.9x + 139.9 \)
\( 5.9x = 264 - 139.9 = 124.1 \)
\( x = \frac{124.1}{5.9} \approx 21 \)
Year = \( 2002 + 21 = 2023 \)

Step 5: Calculate the mean of x and y for the crime data

For the crime data:
\( x \) values: \( 0, 1, 2, 3, 4, 5 \)
\( \bar{x} = \frac{0 + 1 + 2 + 3 + 4 + 5}{6} = \frac{15}{6} = 2.5 \)
\( y \) values: \( 1151, 1131, 1198, 1190, 1251, 1236 \)
\( \bar{y} = \frac{1151 + 1131 + 1198 + 1190 + 1251 + 1236}{6} = \frac{7157}{6} \approx 1192.83 \)

Step 6: Calculate the slope (m) for the crime data

\( (x_0 - \bar{x})(y_0 - \bar{y}) = (0 - 2.5)(1151 - 1192.83) = (-2.5)(-41.83) = 104.575 \)
\( (x_1 - \bar{x})(y_1 - \bar{y}) = (1 - 2.5)(1131 - 1192.83) = (-1.5)(-61.83) = 92.745 \)
\( (x_2 - \bar{x})(y_2 - \bar{y}) = (2 - 2.5)(1198 - 1192.83) = (-0.5)(5.17) = -2.585 \)
\( (x_3 - \bar{x})(y_3 - \bar{y}) = (3 - 2.5)(1190 - 1192.83) = (0.5)(-2.83) = -1.415 \)
\( (x_4 - \bar{x})(y_4 - \bar{y}) = (4 - 2.5)(1251 - 1192.83) = (1.5)(58.17) = 87.255 \)
\( (x_5 - \bar{x})(y_5 - \bar{y}) = (5 - 2.5)(1236 - 1192.83) = (2.5)(43.17) = 107.925 \)
\( \sum (x_i - \bar{x})(y_i - \bar{y}) = 104.575 + 92.745 - 2.585 - 1.415 + 87.255 + 107.925 = 388.4 \)
\( (x_0 - \bar{x})^2 = (0 - 2.5)^2 = 6.25 \)
\( (x_1 - \bar{x})^2 = (1 - 2.5)^2 = 2.25 \)
\( (x_2 - \bar{x})^2 = (2 - 2.5)^2 = 0.25 \)
\( (x_3 - \bar{x})^2 = (3 - 2.5)^2 = 0.25 \)
\( (x_4 - \bar{x})^2 = (4 - 2.5)^2 = 2.25 \)
\( (x_5 - \bar{x})^2 = (5 - 2.5)^2 = 6.25 \)
\( \sum (x_i - \bar{x})^2 = 6.25 + 2.25 + 0.25 + 0.25 + 2.25 + 6.25 = 17.5 \)
\( m = \frac{388.4}{17.5} \approx 22.2 \) (rounded to the nearest tenth)

Step 7: Calculate the y-intercept (b) for the crime data

\( b = \bar{y} - m\bar{x} = 1192.83 - 22.2(2.5) = 1192.83 - 55.5 = 1137.3 \) (rounded to the nearest tenth)
So the linear regression equation for crime is \( y = 22.2x + 1137.3 \)

Step 8: Find the projected number of new cases for 2021

2021 is \( 2021 - 2010 = 11 \) years since 2010, so \( x = 11 \)
\( y = 22.2(11) + 1137.3 = 244.2 + 1137.3 = 1381.5 \approx 1382 \) (rounded to the nearest whole number)

Answer:

• Linear Regression Equation (Profit): \( y = 5.9x + 139.9 \)
• Final Answer (Profit Year): 2023
• Linear Regression Equation (Crime): \( y = 22.2x + 1137.3 \)
• Final Answer (Crime Cases): 1382