Sales of a video game released in the year 2000 took off at first, but then steadily slowed as time moved on. The table below shows the number of games sold, in thousands, from the years 2000–2010.
Year 2000 2001 2002 2003 2004 2005
Number Sold (thousands) 142 149 154 155 159 161
Year 2006 2007 2008 2009 2010 —
Number Sold (thousands) 163 164 164 166 167 —
A. Let x represent the time in years starting with x=1 for the year 2000. Let y represent the number of games sold in thousands.
I got log y=141.91242949 + 10.45366573In(x)
D. If you were to manipulate the data you could do the opposite regression of what you did in a) what is this regression formula?
E. Using the regression formula obtained in d) how many games will be sold in 2015? What did you do to the data from the table to obtain this second regression formula?
D. To obtain the opposite regression formula, you would take the exponential function of the equation you had in part A. So, it would be something like:
y = e^(141.91242949 + 10.45366573x)
E. To obtain the second regression formula, you would need to extrapolate the data beyond the given years and estimate the number of games sold in 2015. Since the data only goes up until 2010, I can't predict the exact number of games sold in 2015. However, using the regression formula, you can plug in x=16 (since x=1 represents the year 2000) and calculate an estimated number of games sold in 2015. Just make sure to account for the fact that x=1 does not represent the year 2000 in this formula.
D. The opposite regression of what you did in part A would involve solving for x in terms of y. To do this, we will rearrange the equation in part A.
Starting with the equation log y = 141.91242949 + 10.45366573 ln(x), we can begin by subtracting 141.91242949 from both sides to isolate the ln(x) term:
log y - 141.91242949 = 10.45366573 ln(x)
Next, dividing both sides by 10.45366573 will give us:
(ln(x)) = (log y - 141.91242949) / 10.45366573
Now, to eliminate the natural logarithm, we can apply the inverse function of a natural logarithm, which is the exponential function e^x:
x = e^[(log y - 141.91242949) / 10.45366573]
Therefore, the opposite regression formula is x = e^[(log y - 141.91242949) / 10.45366573].
E. To estimate the number of games sold in 2015 using the regression formula from part D, we substitute x = 16 for the year 2015. Plugging this value into the formula:
y = e^[(log y - 141.91242949) / 10.45366573]
y = e^[(log y - 141.91242949) / 10.45366573]
y = e^[(log y - 141.91242949) / 10.45366573]
Using a numerical method or a graphing calculator, we can solve for y, which represents the number of games sold in thousands.