Question

11.7 find the equation of a regression line wjc
find the equation for the least squares regression line of the data described below.
a sleep specialist wants to know if meditating before bedtime can help people fall
a sleep. in a study, she asked the subjects to meditate for various lengths of time
just before going to bed.
the specialist tracked the subjects meditation times, x, and how long it took
them to fall asleep, y. both times were recorded in minutes.

round your answers to the nearest thousandth.
( y = square x + square )

Explanation:

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

\(\bar{x}=\frac{1 + 13+52+54+57}{5}=\frac{177}{5} = 35.4\)
\(\bar{y}=\frac{20+30 + 4+11+5}{5}=\frac{70}{5}=14\)

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

The formula for \(b=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}\)

\((x_1-\bar{x})(y_1 - \bar{y})=(1 - 35.4)(20 - 14)=(- 34.4)\times6=-206.4\)
\((x_2-\bar{x})(y_2 - \bar{y})=(13 - 35.4)(30 - 14)=(-22.4)\times16=-358.4\)
\((x_3-\bar{x})(y_3 - \bar{y})=(52 - 35.4)(4 - 14)=(16.6)\times(-10)=-166\)
\((x_4-\bar{x})(y_4 - \bar{y})=(54 - 35.4)(11 - 14)=(18.6)\times(-3)=-55.8\)
\((x_5-\bar{x})(y_5 - \bar{y})=(57 - 35.4)(5 - 14)=(21.6)\times(-9)=-194.4\)

\(\sum_{i = 1}^{5}(x_{i}-\bar{x})(y_{i}-\bar{y})=-206.4-358.4-166-55.8 - 194.4=-981\)

\((x_1-\bar{x})^2=(1 - 35.4)^2=(-34.4)^2 = 1183.36\)
\((x_2-\bar{x})^2=(13 - 35.4)^2=(-22.4)^2 = 501.76\)
\((x_3-\bar{x})^2=(52 - 35.4)^2=(16.6)^2 = 275.56\)
\((x_4-\bar{x})^2=(54 - 35.4)^2=(18.6)^2 = 345.96\)
\((x_5-\bar{x})^2=(57 - 35.4)^2=(21.6)^2 = 466.56\)

\(\sum_{i = 1}^{5}(x_{i}-\bar{x})^{2}=1183.36 + 501.76+275.56+345.96+466.56=2773.2\)

\(b=\frac{-981}{2773.2}\approx - 0.354\)

Step3: Calculate the intercept \(a\)

The formula for \(a=\bar{y}-b\bar{x}\)
\(a = 14-(-0.354)\times35.4=14 + 12.532=26.532\)

Answer:

\(y=-0.354x + 26.532\)