Question

1. the table below shows the recorded temperature on a certain day starting at 6:00 a.m.

time	6:00	6:45	7:30	8:30	9:00	9:45	10:30	11:00
temp	65°	67°	68°	72°	74°	75°	77°	80°

a) find the regression equation.
b) predict the temperature at 2:00 p.m.

Explanation:

Step1: Let \(x\) be the number of hours since 6:00 a.m. and \(y\) be the temperature. Calculate the necessary sums.

Let \(n = 8\) (number of data - points).
\(\sum_{i = 1}^{n}x_{i}=0 + 0.75+1.5 + 2.5+3+3.75+4.5+5=21\)
\(\sum_{i = 1}^{n}y_{i}=65 + 67+68+72+74+75+77+80 = 578\)
\(\sum_{i = 1}^{n}x_{i}^{2}=0^{2}+0.75^{2}+1.5^{2}+2.5^{2}+3^{2}+3.75^{2}+4.5^{2}+5^{2}\)
\(=0 + 0.5625+2.25+6.25+9+14.0625+20.25+25 = 77.375\)
\(\sum_{i = 1}^{n}x_{i}y_{i}=0\times65+0.75\times67 + 1.5\times68+2.5\times72+3\times74+3.75\times75+4.5\times77+5\times80\)
\(=0 + 50.25+102+180+222+281.25+346.5+400 = 1582\)

Step2: Calculate the slope \(m\) of the regression line.

The formula for the slope \(m\) is \(m=\frac{n\sum_{i = 1}^{n}x_{i}y_{i}-\sum_{i = 1}^{n}x_{i}\sum_{i = 1}^{n}y_{i}}{n\sum_{i = 1}^{n}x_{i}^{2}-(\sum_{i = 1}^{n}x_{i})^{2}}\)
Substitute \(n = 8\), \(\sum_{i = 1}^{n}x_{i}=21\), \(\sum_{i = 1}^{n}y_{i}=578\), \(\sum_{i = 1}^{n}x_{i}^{2}=77.375\), and \(\sum_{i = 1}^{n}x_{i}y_{i}=1582\) into the formula:
\[

$$\begin{align*} m&=\frac{8\times1582-21\times578}{8\times77.375 - 21^{2}}\\ &=\frac{12656-12138}{619 - 441}\\ &=\frac{518}{178}\\ & = 2.91 \end{align*}$$

\]

Step3: Calculate the y - intercept \(b\) of the regression line.

The formula for the y - intercept \(b\) is \(b=\frac{\sum_{i = 1}^{n}y_{i}-m\sum_{i = 1}^{n}x_{i}}{n}\)
Substitute \(n = 8\), \(m = 2.91\), \(\sum_{i = 1}^{n}x_{i}=21\), and \(\sum_{i = 1}^{n}y_{i}=578\) into the formula:
\[

$$\begin{align*} b&=\frac{578-2.91\times21}{8}\\ &=\frac{578 - 61.11}{8}\\ &=\frac{516.89}{8}\\ &=64.61 \end{align*}$$

\]

The regression equation is \(y = 2.91x+64.61\)

Step4: Predict the temperature at 2:00 p.m.

Since 2:00 p.m. is 8 hours after 6:00 a.m., substitute \(x = 8\) into the regression equation \(y = 2.91x+64.61\)
\(y=2.91\times8 + 64.61=23.28+64.61 = 87.89\approx87.9^{\circ}\)

Answer:

a) \(y = 2.91x+64.61\)
b) \(87.9^{\circ}\)