Question

is the magnitude of an earthquake related to the depth below the su be the depth (in kilometers) of the quake below the surface at the epicenter. suppose a random sample of earthquakes gave the following information.

x	2.5	4	3.4	4.4	2.4
y	5.2	10.3	10.8	10.3	8.3

draw the line that best fits the data whose scatter diagram is given below.

Explanation:

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

Let \(x_1 = 2.5,x_2=4,x_3 = 3.4,x_4=4.4,x_5 = 2.4\) and \(y_1 = 5.2,y_2=10.3,y_3 = 10.8,y_4=10.3,y_5 = 8.3\)
\(\bar{x}=\frac{\sum_{i = 1}^{5}x_i}{n}=\frac{2.5 + 4+3.4 + 4.4+2.4}{5}=\frac{16.7}{5}=3.34\)
\(\bar{y}=\frac{\sum_{i = 1}^{5}y_i}{n}=\frac{5.2+10.3 + 10.8+10.3+8.3}{5}=\frac{44.9}{5}=8.98\)

Step2: Calculate the slope \(b_1\)

\(b_1=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})(y_i - \bar{y})}{\sum_{i=1}^{n}(x_i-\bar{x})^2}\)
\((x_1-\bar{x})(y_1 - \bar{y})=(2.5 - 3.34)(5.2-8.98)=(- 0.84)\times(-3.78) = 3.1752\)
\((x_2-\bar{x})(y_2 - \bar{y})=(4 - 3.34)(10.3 - 8.98)=0.66\times1.32 = 0.8712\)
\((x_3-\bar{x})(y_3 - \bar{y})=(3.4 - 3.34)(10.8 - 8.98)=0.06\times1.82 = 0.1092\)
\((x_4-\bar{x})(y_4 - \bar{y})=(4.4 - 3.34)(10.3 - 8.98)=1.06\times1.32 = 1.4092\)
\((x_5-\bar{x})(y_5 - \bar{y})=(2.4 - 3.34)(8.3 - 8.98)=(-0.94)\times(-0.68)=0.6392\)
\(\sum_{i = 1}^{5}(x_i-\bar{x})(y_i - \bar{y})=3.1752+0.8712 + 0.1092+1.4092+0.6392=6.204\)
\((x_1-\bar{x})^2=(2.5 - 3.34)^2=(-0.84)^2 = 0.7056\)
\((x_2-\bar{x})^2=(4 - 3.34)^2=0.66^2 = 0.4356\)
\((x_3-\bar{x})^2=(3.4 - 3.34)^2=0.06^2 = 0.0036\)
\((x_4-\bar{x})^2=(4.4 - 3.34)^2=1.06^2 = 1.1236\)
\((x_5-\bar{x})^2=(2.4 - 3.34)^2=(-0.94)^2 = 0.8836\)
\(\sum_{i=1}^{5}(x_i-\bar{x})^2=0.7056+0.4356+0.0036+1.1236+0.8836=3.152\)
\(b_1=\frac{6.204}{3.152}=1.9683\)

Step3: Calculate the y - intercept \(b_0\)

\(b_0=\bar{y}-b_1\bar{x}=8.98-1.9683\times3.34=8.98 - 6.5741=2.4059\)
The equation of the least - squares regression line is \(\hat{y}=b_0 + b_1x=2.4059+1.9683x\)
To draw the line, you can use two points. For \(x = 2\), \(\hat{y}=2.4059+1.9683\times2=2.4059 + 3.9366=6.3425\)
For \(x = 5\), \(\hat{y}=2.4059+1.9683\times5=2.4059+9.8415 = 12.2474\)
Plot the points \((2,6.3425)\) and \((5,12.2474)\) and draw a straight line through them.

Answer:

The equation of the best - fit line is \(\hat{y}=2.4059 + 1.9683x\) and it can be drawn using points such as \((2,6.3425)\) and \((5,12.2474)\)