32 the table below shows the amount of money a popular movie earned, in millions of dollars, during its…

32 the table below shows the amount of money a popular movie earned, in millions of dollars, during its first six weeks in theaters.\n| week (x) | 1 | 2 | 3 | 4 | 5 | 6 |\n|--|--|--|--|--|--|--|\n| dollars earned, in millions (y) | 185 | 150 | 90 | 50 | 25 | 5 |\nwrite the linear regression equation for this data set, rounding all values to the nearest hundredth.\nstate the correlation coefficient to the nearest hundredth.\nstate what this correlation coefficient indicates about the linear fit of the data.

32 the table below shows the amount of money a popular movie earned, in millions of dollars, during its first six weeks in theaters.\n| week (x) | 1 | 2 | 3 | 4 | 5 | 6 |\n|--|--|--|--|--|--|--|\n| dollars earned, in millions (y) | 185 | 150 | 90 | 50 | 25 | 5 |\nwrite the linear regression equation for this data set, rounding all values to the nearest hundredth.\nstate the correlation coefficient to the nearest hundredth.\nstate what this correlation coefficient indicates about the linear fit of the data.

Answer

Explanation:

Step1: Calculate means

Let (x_i) be the week - number and (y_i) be the dollars earned. (n = 6) (\bar{x}=\frac{1 + 2+3+4+5+6}{6}=\frac{21}{6}=3.5) (\bar{y}=\frac{185 + 150+90+50+25+5}{6}=\frac{505}{6}\approx84.17)

Step2: Calculate numerator and denominator for slope (m)

(\sum_{i = 1}^{n}(x_i-\bar{x})(y_i - \bar{y})=(1 - 3.5)(185 - 84.17)+(2 - 3.5)(150 - 84.17)+(3 - 3.5)(90 - 84.17)+(4 - 3.5)(50 - 84.17)+(5 - 3.5)(25 - 84.17)+(6 - 3.5)(5 - 84.17)) (=(- 2.5)\times100.83+(-1.5)\times65.83+(-0.5)\times5.83 + 0.5\times(-34.17)+1.5\times(-59.17)+2.5\times(-79.17)) (=-252.075-98.745 - 2.915-17.085 - 88.755-197.925=-657.5) (\sum_{i = 1}^{n}(x_i-\bar{x})^2=(1 - 3.5)^2+(2 - 3.5)^2+(3 - 3.5)^2+(4 - 3.5)^2+(5 - 3.5)^2+(6 - 3.5)^2) (=(-2.5)^2+(-1.5)^2+(-0.5)^2+0.5^2+1.5^2+2.5^2) (=6.25 + 2.25+0.25+0.25+2.25+6.25 = 17.5) (m=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})(y_i - \bar{y})}{\sum_{i = 1}^{n}(x_i-\bar{x})^2}=\frac{-657.5}{17.5}\approx - 37.57)

Step3: Calculate (y) - intercept (b)

(b=\bar{y}-m\bar{x}) (b = 84.17-(-37.57)\times3.5) (b = 84.17+131.495=215.67) The linear regression equation is (y=-37.57x + 215.67)

Step4: Calculate correlation coefficient (r)

(s_x=\sqrt{\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n - 1}}=\sqrt{\frac{17.5}{5}}=\sqrt{3.5}\approx1.87) (s_y=\sqrt{\frac{\sum_{i = 1}^{n}(y_i-\bar{y})^2}{n - 1}}) (\sum_{i = 1}^{n}(y_i-\bar{y})^2=(185 - 84.17)^2+(150 - 84.17)^2+(90 - 84.17)^2+(50 - 84.17)^2+(25 - 84.17)^2+(5 - 84.17)^2) (=100.83^2+65.83^2+5.83^2+(-34.17)^2+(-59.17)^2+(-79.17)^2) (=10166.99+4333.59+34.09+1167.59+3501.09+6267.99 = 25471.34) (s_y=\sqrt{\frac{25471.34}{5}}\approx71.37) (r=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})(y_i - \bar{y})}{(n - 1)s_xs_y}=\frac{-657.5}{5\times1.87\times71.37}=\frac{-657.5}{671.97}\approx - 0.98)

Step5: Interpret correlation coefficient

Since (r\approx - 0.98), which is close to - 1, it indicates a strong negative linear relationship between the week number and the dollars earned.

Answer:

Linear regression equation: (y=-37.57x + 215.67) Correlation coefficient: (r\approx - 0.98) Interpretation: There is a strong negative linear relationship between the week number and the dollars earned.