find the equation for the least - squares regression line of the data described below. a gym franchise was…

find the equation for the least - squares regression line of the data described below. a gym franchise was considering a television marketing campaign to increase its membership. the franchises market researchers wanted to get a better sense of the television and exercise habits of the gyms target demographic. to begin, the market researchers surveyed some of the current members about how many hours they had spent watching television and exercising last month. using the survey responses, the researchers compared the number of hours of television watched, x, to the number of hours of exercise, y, for each member.\nhours of television\thours of exercise\n24.06\t55.72\n17.80\t32.18\n13.96\t50.13\n9.01\t58.53\n7.34\t7.93\n2.02\t8.74\nround your answers to the nearest thousandth.\ny = x +

find the equation for the least - squares regression line of the data described below. a gym franchise was considering a television marketing campaign to increase its membership. the franchises market researchers wanted to get a better sense of the television and exercise habits of the gyms target demographic. to begin, the market researchers surveyed some of the current members about how many hours they had spent watching television and exercising last month. using the survey responses, the researchers compared the number of hours of television watched, x, to the number of hours of exercise, y, for each member.\nhours of television\thours of exercise\n24.06\t55.72\n17.80\t32.18\n13.96\t50.13\n9.01\t58.53\n7.34\t7.93\n2.02\t8.74\nround your answers to the nearest thousandth.\ny = x +

Answer

Explanation:

Step1: Recall least - squares regression formula

The equation of a least - squares regression line is $y = a+bx$, where $b=\frac{n\sum_{i = 1}^{n}x_iy_i-\sum_{i = 1}^{n}x_i\sum_{i = 1}^{n}y_i}{n\sum_{i = 1}^{n}x_i^{2}-(\sum_{i = 1}^{n}x_i)^{2}}$ and $a=\bar{y}-b\bar{x}$, $\bar{x}=\frac{\sum_{i = 1}^{n}x_i}{n}$, $\bar{y}=\frac{\sum_{i = 1}^{n}y_i}{n}$, and $n$ is the number of data points. Let $x$ be the hours of television watched and $y$ be the hours of exercise. Here $n = 6$. First, calculate $\sum_{i = 1}^{6}x_i=2.02 + 7.34+9.01+13.96+17.80+24.06=74.19$. $\sum_{i = 1}^{6}y_i=55.72+32.18+50.13+58.53+7.93+8.74 = 213.23$. $\sum_{i = 1}^{6}x_i^{2}=2.02^{2}+7.34^{2}+9.01^{2}+13.96^{2}+17.80^{2}+24.06^{2}=4.0804 + 53.8756+81.1801+194.8816+316.84+578.8836 = 1229.8419$. $\sum_{i = 1}^{6}x_iy_i=2.02\times55.72+7.34\times32.18+9.01\times50.13+13.96\times58.53+17.80\times7.93+24.06\times8.74=112.5544+236.1012+451.6713+817.0788+141.154+209.2844 = 1967.8441$.

Step2: Calculate $\bar{x}$ and $\bar{y}$

$\bar{x}=\frac{\sum_{i = 1}^{6}x_i}{6}=\frac{74.19}{6}=12.365$. $\bar{y}=\frac{\sum_{i = 1}^{6}y_i}{6}=\frac{213.23}{6}=35.5383$.

Step3: Calculate $b$

$b=\frac{6\times1967.8441 - 74.19\times213.23}{6\times1229.8419-(74.19)^{2}}$ $=\frac{11807.0646-15719.3337}{7379.0514 - 5504.1561}=\frac{- 3912.2691}{1874.8953}\approx - 2.0866$.

Step4: Calculate $a$

$a=\bar{y}-b\bar{x}=35.5383-(-2.0866)\times12.365$ $=35.5383 + 25.7177=61.256$.

Answer:

$y=-2.087x + 61.257$