Question

A study by the Atlanta, Georgia, Department of Transportation on the effect of bus ticket prices on the number of passengers produced the following results:
Ticket price (cents):

25 30 35 40 45 50 55 60

Passengers per 100 miles:

800 780 780 660 640 600 620 620

Plot these data.
Develop the estimating equation that best describes these data.
Predict the number of passengers per 100 miles if the ticket price were 70 cents.

Answer 1

Answer:

a) Figure attached

b) $y=-6.238 x +952.62$

c) $y=-6.238*70 +952.62=515.955$

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 $y =mx+b$

For this case we need to calculate the slope with the following formula:

$m=\frac{S_{xy}}{S_{xx}}$

Where:

$S_{xy}=\sum_{i=1}^n x_i y_i -\frac{(\sum_{i=1}^n x_i)(\sum_{i=1}^n y_i)}{n}$

$S_{xx}=\sum_{i=1}^n x^2_i -\frac{(\sum_{i=1}^n x_i)^2}{n}$

So we can find the sums like this:

$\sum_{i=1}^n x_i =340$

$\sum_{i=1}^n y_i =5500$

$\sum_{i=1}^n x^2_i =15500$

$\sum_{i=1}^n y^2_i =3830800$

$\sum_{i=1}^n x_i y_i =227200$

With these we can find the sums:

$S_{xx}=\sum_{i=1}^n x^2_i -\frac{(\sum_{i=1}^n x_i)^2}{n}=15500-\frac{340^2}{8}=1050$

$S_{xy}=\sum_{i=1}^n x_i y_i -\frac{(\sum_{i=1}^n x_i)(\sum_{i=1}^n y_i)}{n}=227200-\frac{340*5500}{8}=-6550$

And the slope would be:

$m=-\frac{6550}{1050}=-6.238$

Nowe we can find the means for x and y like this:

$\bar x= \frac{\sum x_i}{n}=\frac{340}{8}=42.5$

$\bar y= \frac{\sum y_i}{n}=\frac{5500}{8}=687.5$

And we can find the intercept using this:

$b=\bar y -m \bar x=687.5-(-6.238*42.5)=952.62$

So the line would be given by:

$y=-6.238 x +952.62$

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:

$y=-6.238*70 +952.62=515.955$