Question

payton collected data to show the relationship between the number of hours he practices and the number of errors he makes when playing a new piece of music. the table shows his data. practice makes better which is the approximate slope of the line of best fit for the data? -5.5 -4.5 -2.0 -1.0

Explanation:

Step1: Calculate the mean of \(x\) (number of hours) and \(y\) (number of errors)

Let \(x\) values be \(x_1 = 1,x_2=2,x_3 = 3,x_4=4,x_5 = 5,x_6=6,x_7 = 7,x_8=8\)
Let \(y\) values be \(y_1 = 36,y_2=34,y_3 = 30,y_4=31,y_5 = 23,y_6=16,y_7 = 11,y_8=5\)
The mean of \(x\), \(\bar{x}=\frac{1 + 2+3+4+5+6+7+8}{8}=\frac{36}{8}=4.5\)
The mean of \(y\), \(\bar{y}=\frac{36 + 34+30+31+23+16+11+5}{8}=\frac{186}{8}=23.25\)

Step2: Calculate \(\sum_{i = 1}^{n}(x_i-\bar{x})(y_i - \bar{y})\) and \(\sum_{i=1}^{n}(x_i-\bar{x})^2\)

\((x_1-\bar{x})(y_1-\bar{y})=(1 - 4.5)(36-23.25)=(- 3.5)\times12.75=-44.625\)
\((x_2-\bar{x})(y_2-\bar{y})=(2 - 4.5)(34 - 23.25)=(-2.5)\times10.75=-26.875\)
\((x_3-\bar{x})(y_3-\bar{y})=(3 - 4.5)(30 - 23.25)=(-1.5)\times6.75=-10.125\)
\((x_4-\bar{x})(y_4-\bar{y})=(4 - 4.5)(31 - 23.25)=(-0.5)\times7.75=-3.875\)
\((x_5-\bar{x})(y_5-\bar{y})=(5 - 4.5)(23 - 23.25)=(0.5)\times(-0.25)=-0.125\)
\((x_6-\bar{x})(y_6-\bar{y})=(6 - 4.5)(16 - 23.25)=(1.5)\times(-7.25)=-10.875\)
\((x_7-\bar{x})(y_7-\bar{y})=(7 - 4.5)(11 - 23.25)=(2.5)\times(-12.25)=-30.625\)
\((x_8-\bar{x})(y_8-\bar{y})=(8 - 4.5)(5 - 23.25)=(3.5)\times(-18.25)=-63.875\)
\(\sum_{i = 1}^{8}(x_i-\bar{x})(y_i-\bar{y})=-44.625-26.875-10.125-3.875-0.125-10.875-30.625-63.875=-191.125\)

\((x_1-\bar{x})^2=(1 - 4.5)^2=(-3.5)^2 = 12.25\)
\((x_2-\bar{x})^2=(2 - 4.5)^2=(-2.5)^2 = 6.25\)
\((x_3-\bar{x})^2=(3 - 4.5)^2=(-1.5)^2 = 2.25\)
\((x_4-\bar{x})^2=(4 - 4.5)^2=(-0.5)^2 = 0.25\)
\((x_5-\bar{x})^2=(5 - 4.5)^2=(0.5)^2 = 0.25\)
\((x_6-\bar{x})^2=(6 - 4.5)^2=(1.5)^2 = 2.25\)
\((x_7-\bar{x})^2=(7 - 4.5)^2=(2.5)^2 = 6.25\)
\((x_8-\bar{x})^2=(8 - 4.5)^2=(3.5)^2 = 12.25\)
\(\sum_{i=1}^{8}(x_i-\bar{x})^2=12.25+6.25+2.25+0.25+0.25+2.25+6.25+12.25 = 42\)

Step3: Calculate the slope \(m\)

The formula for the slope of the line of best - fit is \(m=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})(y_i-\bar{y})}{\sum_{i=1}^{n}(x_i-\bar{x})^2}\)
Substitute \(\sum_{i = 1}^{n}(x_i-\bar{x})(y_i-\bar{y})=-191.125\) and \(\sum_{i=1}^{n}(x_i-\bar{x})^2 = 42\) into the formula
\(m=\frac{-191.125}{42}\approx - 4.55\approx-4.5\)

Answer:

-4.5