Question

3-12. mr. saksunn is concerned both about his students’ scores on the last math test, and about the number of students looking tired in class. he decides to see if there is a relationship between the number of tired or sleepy behaviors (yawns, nodding - off, head on desk) a student exhibits and their test score. he has his assistant observe 10 students and count the number of tired behaviors during one week of class.

tired behaviors	2	4	0	2	1	7	0	1	3	6
test score	73	63	89	85	90	58	97	90	79	41

a. make a scatterplot on graph paper and draw the line of best fit. determine the equation of the line of best fit.
b. using your equation from part (a), predict the test score of a student who exhibits 5 tired behaviors during professor saksunn’s math class in a week.
-13. draw a slope triangle and use it to write the equation of the line shown in the graph at right.

Explanation:

Step1: Calculate the means of x (tired behaviors) and y (test scores)

Let \(x = [2,4,0,2,1,7,0,1,3,6]\) and \(y=[73,63,89,85,90,58,97,90,79,41]\)
\(\bar{x}=\frac{2 + 4+0+2+1+7+0+1+3+6}{10}=\frac{26}{10} = 2.6\)
\(\bar{y}=\frac{73 + 63+89+85+90+58+97+90+79+41}{10}=\frac{775}{10}=77.5\)

Step2: Calculate the slope \(m\)

\[m=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}\]
\(\sum_{i = 1}^{10}(x_{i}-\bar{x})(y_{i}-\bar{y})=(2 - 2.6)(73 - 77.5)+(4 - 2.6)(63 - 77.5)+\cdots+(6 - 2.6)(41 - 77.5)\)
\(=( - 0.6)\times(-4.5)+1.4\times(-14.5)+\cdots+3.4\times(-36.5)\)
\(=2.7-20.3+\cdots - 124.1=-319\)
\(\sum_{i = 1}^{10}(x_{i}-\bar{x})^{2}=(2 - 2.6)^{2}+(4 - 2.6)^{2}+\cdots+(6 - 2.6)^{2}\)
\(=(-0.6)^{2}+1.4^{2}+\cdots+3.4^{2}=48.4\)
\(m=\frac{-319}{48.4}=- 6.6\)

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

Using the point - slope form \(y - y_{1}=m(x - x_{1})\) with \((x_{1},y_{1})=(\bar{x},\bar{y})\)
\(b=\bar{y}-m\bar{x}=77.5-(-6.6)\times2.6=77.5 + 17.16=94.66\)
The equation of the line of best fit is \(y=-6.6x + 94.66\)

Step4: Predict the test score for \(x = 5\)

Substitute \(x = 5\) into \(y=-6.6x + 94.66\)
\(y=-6.6\times5+94.66=-33 + 94.66 = 61.66\)

Answer:

a. The equation of the line of best fit is \(y=-6.6x + 94.66\)
b. The predicted test score is \(61.66\)