Respuesta :
Answer:
[tex]y=-1.055 x +37.643[/tex]
And the best option would be:
a. ŷ= 37.643-1.0543x
Step-by-step explanation:
We assume that the data is this one:
x: 34.0, 32.5, 35.0, 31.0, 30.0, 27.5, 29.0
y: 1.30, 3.25, 0.65, 6.50, 5.85, 8.45, 6.50
Find the least-squares line appropriate for this data.
For this case we need to calculate the slope with the following formula:
[tex]m=\frac{S_{xy}}{S_{xx}}[/tex]
Where:
[tex]S_{xy}=\sum_{i=1}^n x_i y_i -\frac{(\sum_{i=1}^n x_i)(\sum_{i=1}^n y_i)}{n}[/tex]
[tex]S_{xx}=\sum_{i=1}^n x^2_i -\frac{(\sum_{i=1}^n x_i)^2}{n}[/tex]
So we can find the sums like this:
[tex]\sum_{i=1}^n x_i = 34.0+ 32.5+ 35.0+ 31.0+ 30.0+ 27.5+ 29.0 =219[/tex]
[tex]\sum_{i=1}^n y_i =1.3+3.25+0.65+6.5+5.85+8.45+6.5=32.5[/tex]
[tex]\sum_{i=1}^n x^2_i = 34.0^2+ 32.5^2+ 35.0^2+ 31.0^2+ 30.0^2+ 27.5^2+ 29.0^2 =6895.5[/tex]
[tex]\sum_{i=1}^n y^2_i =1.3^2+3.25^2+0.65^2+6.5^2+5.85^2+8.45^2+6.5^2=202.8[/tex]
[tex]\sum_{i=1}^n x_i y_i =34*1.3 + 32.5*3.25+ 35*0.65+ 31*6.5+30*5.585 +27.5*8.45 +29*6.5=970.45[/tex]
With these we can find the sums:
[tex]S_{xx}=\sum_{i=1}^n x^2_i -\frac{(\sum_{i=1}^n x_i)^2}{n}=6895.5-\frac{219^2}{7}=43.929[/tex]
[tex]S_{xy}=\sum_{i=1}^n x_i y_i -\frac{(\sum_{i=1}^n x_i)(\sum_{i=1}^n y_i)}{n}=970.45-\frac{219*32.5}{7}=-46.336[/tex]
And the slope would be:
[tex]m=-\frac{46.336}{43.929}=-1.055[/tex]
Nowe we can find the means for x and y like this:
[tex]\bar x= \frac{\sum x_i}{n}=\frac{219}{7}=31.286[/tex]
[tex]\bar y= \frac{\sum y_i}{n}=\frac{32.5}{7}=4.643[/tex]
And we can find the intercept using this:
[tex]b=\bar y -m \bar x=4.643-(-1.055*31.286)=37.643[/tex]
So the line would be given by:
[tex]y=-1.055 x +37.643[/tex]
And the best option would be:
a. ŷ= 37.643-1.0543x
