Question

the table below shows the number of hours ten students spent studying for a test and their scores.
write the linear regression equation for this data set. round all values to the nearest tenth.
hours spent studying (x) test scores (y)
0 35
1 40
2 46
4 65
4 67
4 70
6 82
6 88
7 82
8 95

Explanation:

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

First, find the sum of \(x\) values: \(0 + 1+2 + 4+4+4+6+6+7+8=42\). The mean of \(x\), \(\bar{x}=\frac{42}{10} = 4.2\).
Find the sum of \(y\) values: \(35+40 + 46+65+67+70+82+88+82+95 = 670\). The mean of \(y\), \(\bar{y}=\frac{670}{10}=67\).

Step2: Calculate the slope \(b\)

The formula for \(b\) is \(b=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}\).
Calculate \((x_{i}-\bar{x})(y_{i}-\bar{y})\) for each pair:
\((0 - 4.2)(35 - 67)=(- 4.2)\times(-32)=134.4\)
\((1 - 4.2)(40 - 67)=(-3.2)\times(-27)=86.4\)
\((2 - 4.2)(46 - 67)=(-2.2)\times(-21)=46.2\)
\((4 - 4.2)(65 - 67)=(-0.2)\times(-2)=0.4\)
\((4 - 4.2)(67 - 67)=(-0.2)\times0 = 0\)
\((4 - 4.2)(70 - 67)=(-0.2)\times3=-0.6\)
\((6 - 4.2)(82 - 67)=(1.8)\times15 = 27\)
\((6 - 4.2)(88 - 67)=(1.8)\times21=37.8\)
\((7 - 4.2)(82 - 67)=(2.8)\times15 = 42\)
\((8 - 4.2)(95 - 67)=(3.8)\times28 = 106.4\)
Sum of \((x_{i}-\bar{x})(y_{i}-\bar{y})\): \(134.4+86.4 + 46.2+0.4+0-0.6+27+37.8+42+106.4=470\)
Calculate \((x_{i}-\bar{x})^{2}\) for each \(x_{i}\):
\((0 - 4.2)^{2}=17.64\)
\((1 - 4.2)^{2}=10.24\)
\((2 - 4.2)^{2}=4.84\)
\((4 - 4.2)^{2}=0.04\)
\((4 - 4.2)^{2}=0.04\)
\((4 - 4.2)^{2}=0.04\)
\((6 - 4.2)^{2}=3.24\)
\((6 - 4.2)^{2}=3.24\)
\((7 - 4.2)^{2}=7.84\)
\((8 - 4.2)^{2}=14.44\)
Sum of \((x_{i}-\bar{x})^{2}\): \(17.64+10.24+4.84+0.04+0.04+0.04+3.24+3.24+7.84+14.44 = 51.6\)
\(b=\frac{470}{51.6}\approx9.1\)

Step3: Calculate the intercept \(a\)

The formula for \(a\) is \(a=\bar{y}-b\bar{x}\).
\(a = 67-9.1\times4.2=67 - 38.22=28.8\)

Answer:

The linear regression equation is \(y = 9.1x+28.8\)