Question

find the equation for the least squares regression line of the data described below. audrey has trouble getting her math homework done on time, and her mother suspects it is due to lack of sleep. for the next few nights, audreys mother notes the number of hours she sleeps, x, and the number of minutes it takes her to do her math homework the following day, y. hours slept minutes needed to finish math homework 5.28 168 5.59 180 5.65 151 5.73 175 7.48 168 8.70 125 8.86 112 round your answers to the nearest thousandth. y = x + submit

Explanation:

Step1: Calculate the means of \(x\) and \(y\)

Let \(x_i\) be the hours - slept values and \(y_i\) be the minutes - needed values.
\(n = 7\)
\(\bar{x}=\frac{\sum_{i = 1}^{n}x_i}{n}=\frac{5.28 + 5.59+5.65 + 5.73+7.48+8.70+8.86}{7}=\frac{47.29}{7}\approx6.756\)
\(\bar{y}=\frac{\sum_{i = 1}^{n}y_i}{n}=\frac{168 + 180+151+175+168+125+112}{7}=\frac{1079}{7}\approx154.143\)

Step2: Calculate the numerator and denominator for the slope \(b_1\)

\(\sum_{i = 1}^{n}(x_i-\bar{x})(y_i - \bar{y})=(5.28 - 6.756)(168 - 154.143)+(5.59 - 6.756)(180 - 154.143)+(5.65 - 6.756)(151 - 154.143)+(5.73 - 6.756)(175 - 154.143)+(7.48 - 6.756)(168 - 154.143)+(8.70 - 6.756)(125 - 154.143)+(8.86 - 6.756)(112 - 154.143)\)
\(=(- 1.476)\times13.857+(-1.166)\times25.857+(-1.106)\times(-3.143)+(-1.026)\times20.857+(0.724)\times13.857+(1.944)\times(-29.143)+(2.104)\times(-42.143)\)
\(\approx-20.453-30.049 + 3.476-21.399+10.043-56.654 - 88.679=-203.315\)

\(\sum_{i = 1}^{n}(x_i-\bar{x})^2=(5.28 - 6.756)^2+(5.59 - 6.756)^2+(5.65 - 6.756)^2+(5.73 - 6.756)^2+(7.48 - 6.756)^2+(8.70 - 6.756)^2+(8.86 - 6.756)^2\)
\(=(-1.476)^2+(-1.166)^2+(-1.106)^2+(-1.026)^2+(0.724)^2+(1.944)^2+(2.104)^2\)
\(=2.179+1.360+1.223+1.052+0.524+3.779+4.427 = 14.544\)

\(b_1=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})(y_i - \bar{y})}{\sum_{i = 1}^{n}(x_i-\bar{x})^2}=\frac{-203.315}{14.544}\approx - 13.980\)

Step3: Calculate the y - intercept \(b_0\)

\(b_0=\bar{y}-b_1\bar{x}\)
\(b_0 = 154.143-(-13.980)\times6.756\)
\(b_0=154.143 + 94.459=248.602\)

Answer:

\(y=-13.980x + 248.602\)