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 —
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? In a) I used logarithmic and got y=141.91242949 + 10.45366573
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?
To obtain the opposite regression formula, you need to reverse the process from step a), where you used a logarithmic regression. In this case, you need to use the inverse of the logarithmic function, which is an exponential function.
To find the exponential regression formula, we can use the exponential regression formula:
y = ab^x
Where:
- y represents the number of games sold (in thousands).
- a represents the value of y when x is 0.
- b represents the exponential growth rate.
To determine the values of a and b, we can use the data from the table. We will use x as the years from 2000 onwards.
Using the provided data, we can create a new table as follows:
Year | Number Sold (in thousands)
-----------------------------------
2000 | 142
2001 | 149
2002 | 154
2003 | 155
2004 | 159
2005 | 161
2006 | 163
2007 | 164
2008 | 164
2009 | 166
2010 | 167
To transform this data into exponential form, we need to take the logarithm of the sales figures.
Year | Log of Number Sold
-----------------------------------
2000 | log(142)
2001 | log(149)
2002 | log(154)
2003 | log(155)
2004 | log(159)
2005 | log(161)
2006 | log(163)
2007 | log(164)
2008 | log(164)
2009 | log(166)
2010 | log(167)
After calculating the logarithm of the number of games sold, you can perform a linear regression with the transformed data.
The resulting regression formula in exponential form will be of the form:
y = a * e^(bx)
Where:
- y represents the transformed logarithmic values of the number of games sold.
- a represents the value of y when x is 0.
- b represents the growth rate.
By solving the linear regression equation with this exponential form, you will obtain the exponential regression formula.
To calculate the number of games that will be sold in 2015 using this regression formula, substitute x = 15 (representing the year 2015) into the equation and solve for y. The value of y will represent the number of games sold in thousands.