Question

in a mathematics class of ten students, the teacher wanted to determine how a homework grade influenced a student’s performance on the subsequent test. the homework grade and subsequent test grade for each student are given in the accompanying table. table: homework grade (x) values: 94, 95, 92, 87, 82, 80, 75, 65, 50, 30; test grade (y) values: 99, 94, 95, 89, 86, 78, 73, 67, 45, 40 what is the y-intercept of this equation? round your answer to the nearest hundredth.

Explanation:

To find the y - intercept of the linear regression equation \(y = mx + b\) (where \(b\) is the y - intercept), we first need to calculate the slope \(m\) and then use the formula \(b=\bar{y}-m\bar{x}\), where \(\bar{x}\) is the mean of the \(x\) (homework grade) values and \(\bar{y}\) is the mean of the \(y\) (test grade) values.

Step 1: Calculate \(\bar{x}\) and \(\bar{y}\)

First, we list out the \(x\) (homework grade) values: \(94, 95, 92, 87, 82, 90, 75, 65, 50, 90\)
The sum of \(x\) values, \(\sum x=94 + 95+92 + 87+82+90+75+65+50+90\)
\[

$$\begin{align*} \sum x&=(94 + 95)+(92 + 87)+(82 + 90)+(75 + 65)+(50 + 90)\\ &=189+179+172+140+140\\ &=189+179 = 368; 368+172=540; 540 + 140=680; 680+140 = 820 \end{align*}$$

\]
Since there are \(n = 10\) data points, \(\bar{x}=\frac{\sum x}{n}=\frac{820}{10}=82\)

Now, the \(y\) (test grade) values: \(99, 94, 95, 89, 86, 78, 73, 67, 45, 40\)
The sum of \(y\) values, \(\sum y=99+94 + 95+89+86+78+73+67+45+40\)
\[

$$\begin{align*} \sum y&=(99 + 94)+(95 + 89)+(86 + 78)+(73 + 67)+(45 + 40)\\ &=193+184+164+140+85\\ &=193+184 = 377; 377+164=541; 541+140 = 681; 681+85=766 \end{align*}$$

\]
\(\bar{y}=\frac{\sum y}{n}=\frac{766}{10} = 76.6\)

Step 2: Calculate the slope \(m\)

The formula for the slope \(m\) of the linear regression line is \(m=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}\)

First, we calculate \((x_{i}-\bar{x})\) and \((y_{i}-\bar{y})\) for each data point:

\(x_i\)	\(y_i\)	\(x_i-\bar{x}\)	\(y_i - \bar{y}\)	\((x_i-\bar{x})(y_i - \bar{y})\)	\((x_i-\bar{x})^2\)
95	94	\(95 - 82 = 13\)	\(94 - 76.6=17.4\)	\(13\times17.4 = 226.2\)	\(13^2 = 169\)
92	95	\(92 - 82=10\)	\(95 - 76.6 = 18.4\)	\(10\times18.4=184\)	\(10^2 = 100\)
87	89	\(87 - 82 = 5\)	\(89 - 76.6=12.4\)	\(5\times12.4 = 62\)	\(5^2=25\)
82	86	\(82 - 82=0\)	\(86 - 76.6 = 9.4\)	\(0\times9.4 = 0\)	\(0^2=0\)
90	78	\(90 - 82 = 8\)	\(78 - 76.6=1.4\)	\(8\times1.4 = 11.2\)	\(8^2=64\)
75	73	\(75 - 82=-7\)	\(73 - 76.6=-3.6\)	\((-7)\times(-3.6)=25.2\)	\((-7)^2 = 49\)
65	67	\(65 - 82=-17\)	\(67 - 76.6=-9.6\)	\((-17)\times(-9.6)=163.2\)	\((-17)^2 = 289\)
50	45	\(50 - 82=-32\)	\(45 - 76.6=-31.6\)	\((-32)\times(-31.6)=1011.2\)	\((-32)^2=1024\)
90	40	\(90 - 82 = 8\)	\(40 - 76.6=-36.6\)	\(8\times(-36.6)=-292.8\)	\(8^2 = 64\)

Now, calculate \(\sum(x_{i}-\bar{x})(y_{i}-\bar{y})\):
\[

$$\begin{align*} \sum(x_{i}-\bar{x})(y_{i}-\bar{y})&=268.8+226.2 + 184+62+0+11.2+25.2+163.2+1011.2-292.8\\ &=(268.8+226.2)+(184 + 62)+(0+11.2)+(25.2+163.2)+(1011.2-292.8)\\ &=495+246+11.2+188.4+718.4\\ &=495+246=741; 741+11.2 = 752.2; 752.2+188.4=940.6; 940.6+718.4 = 1659 \end{align*}$$

\]

Calculate \(\sum(x_{i}-\bar{x})^2\):
\[

$$\begin{align*} \sum(x_{i}-\bar{x})^2&=144+169+100+25+0+64+49+289+1024+64\\ &=(144+169)+(100+25)+(0+64)+(49+289)+(1024+64)\\ &=313+125+64+338+1088\\ &=313+125 = 438; 438+64=502; 502+338=840; 840+1088=1928 \end{align*}$$

\]

So, the slope \(m=\frac{1659}{1928}\approx0.8605\)

Step 3: Calculate the y - intercept \(b\)

We use the formula \(b=\bar{y}-m\bar{x}\)
We know that \(\bar{y} = 76.6\), \(m\approx0.8605\) and \(\bar{x}=82\)
\[

$$\begin{align*} b&=76.6-0.8605\times82\\ &=76.6 - 70.561\\ &=6.039\approx6.04 \end{align*}$$

\]

Answer:

\(6.04\)