Average gas prices

Question

Columbia, SC
Year Gas Price
2002 $0.70
2003 $1.04
2004 $1.15
2005 $1.38
2006 $1.86
2007 $1.70
2008 $2.55
2009 $1.29
2010 $2.22
2011 $2.56
Cabs use a valuable commodity-gas!
List the range of the linear regression equation in interval notation. Round to the nearest thousandth if needed

Answer 1

To estimate the range of the linear regression equation given the data, it's important to first calculate the linear regression line. However, I can't directly perform complex calculations like fitting a linear regression here, but I can guide you on the process.

If you have access to statistical software or a calculator, you can input the data and calculate the linear regression model. If you are doing it manually, you can use the following formulas to calculate the slope (m) and y-intercept (b) of the linear regression line (y = mx + b):

1. Calculate the slope (m) of the line:
\( m = \frac{n(Σxy) - (Σx)(Σy)}{n(Σx^2) - (Σx)^2} \)

where n is the number of data points, Σxy is the sum of the product of corresponding x and y values, Σx is the sum of x values, Σy is the sum of y values, and Σx^2 is the sum of squared x values.

2. Calculate the y-intercept (b) of the line:
\( b = \frac{Σy - m(Σx)}{n} \)

After calculating m and b, plug in the min and max x values (which represent the years in your dataset) to find the corresponding y values (which represent the gas prices). The min x value is 2002 and the max x value is 2011.

Let's say that after you have calculated m and b, and plugged in the x values, you received two y values: y_min for x=2002 and y_max for x=2011.

Then, the range of the linear regression equation would be the interval from y_min to y_max:

\[ (\text{y_min}, \text{y_max}) \]

Remember, "range" in the context of functions usually refers to the set of all possible output values (the y values), which, given a large enough sample of years, might exceed the specific y values obtained by plugging in the min and max x values. However, for this specific set of years (2002 to 2011), the range could be just as I've indicated.

Make sure to round the calculated y_min and y_max to the nearest thousandth, as per your requirement.