Answer:
a) Figure attached
b) [tex]y=-6.238 x +952.62[/tex]
c) [tex]y=-6.238*70 +952.62=515.955[/tex]
Step-by-step explanation:
For this case we assume that x= "Ticket price (cents)" and y ="assengers per 100 miles"
X: 25 30 35 40 45 50 55 60
Y: 800 780 780 660 640 600 620 620
Plot these data.
And the plot is on the figure attached. We see a linear inverse relationship between the two variables.
Develop the estimating equation that best describes these data.
We want to estimate a linear model [tex] y =mx+b[/tex]
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 =340[/tex]
[tex]\sum_{i=1}^n y_i =5500[/tex]
[tex]\sum_{i=1}^n x^2_i =15500[/tex]
[tex]\sum_{i=1}^n y^2_i =3830800[/tex]
[tex]\sum_{i=1}^n x_i y_i =227200[/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}=15500-\frac{340^2}{8}=1050[/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}=227200-\frac{340*5500}{8}=-6550[/tex]
And the slope would be:
[tex]m=-\frac{6550}{1050}=-6.238[/tex]
Nowe we can find the means for x and y like this:
[tex]\bar x= \frac{\sum x_i}{n}=\frac{340}{8}=42.5[/tex]
[tex]\bar y= \frac{\sum y_i}{n}=\frac{5500}{8}=687.5[/tex]
And we can find the intercept using this:
[tex]b=\bar y -m \bar x=687.5-(-6.238*42.5)=952.62[/tex]
So the line would be given by:
[tex]y=-6.238 x +952.62[/tex]
Predict the number of passengers per 100 miles if the ticket price were 70 cents.
For this case we just need to replace x=70 in our model and we got:
[tex]y=-6.238*70 +952.62=515.955[/tex]
