Question

ben started collecting rare comic books 14 years ago when his uncle gave him a box of old comics. this table shows the value, in dollars, of ben’s collection every two years since he started collecting comic books. bens comic book collection years (x) 2 4 6 8 10 12 14 value (y) 717 790 871 960 1,059 1,167 1,287 use exponential regression to find the function that best fits the data. write your answer in the form $y = a(b)^x$. round a and b to the nearest hundredth. $y = $ use the exponential function to predict how much ben’s collection was worth when he first started collecting comic books. round your answer to the nearest dollar. $

Explanation:

Step1: Recall exponential regression form

The target form is $y = a(b)^x$, which can be linearized by taking the natural log of both sides: $\ln(y) = \ln(a) + x\ln(b)$, where we let $Y=\ln(y)$, $A=\ln(a)$, $B=\ln(b)$, so $Y = A + Bx$.

Step2: Calculate $\ln(y)$ for each data point

$x$	$y$	$\ln(y)$
4	790	$\ln(790)\approx6.673$
6	871	$\ln(871)\approx6.771$
8	960	$\ln(960)\approx6.867$
10	1059	$\ln(1059)\approx6.964$
12	1167	$\ln(1167)\approx7.062$
14	1287	$\ln(1287)\approx7.159$

Step3: Compute linear regression for $Y$ and $x$

First, calculate necessary sums:

• $n=7$
• $\sum x = 2+4+6+8+10+12+14=56$
• $\sum Y = 6.575+6.673+6.771+6.867+6.964+7.062+7.159\approx48.071$
• $\sum xY = (2*6.575)+(4*6.673)+(6*6.771)+(8*6.867)+(10*6.964)+(12*7.062)+(14*7.159)\approx390.232$
• $\sum x^2 = 2^2+4^2+6^2+8^2+10^2+12^2+14^2=560$

Calculate slope $B$:

$$ B = \frac{n\sum xY - \sum x \sum Y}{n\sum x^2 - (\sum x)^2} = \frac{7*390.232 - 56*48.071}{7*560 - 56^2} \approx \frac{2731.624 - 2691.976}{3920 - 3136} = \frac{39.648}{784} \approx 0.0506 $$

Calculate intercept $A$:

$$ A = \frac{\sum Y - B\sum x}{n} = \frac{48.071 - 0.0506*56}{7} \approx \frac{48.071 - 2.834}{7} \approx \frac{45.237}{7} \approx 6.462 $$

Step4: Solve for $a$ and $b$

$b = e^B \approx e^{0.0506} \approx 1.05$
$a = e^A \approx e^{6.462} \approx 643.00$
So the exponential function is $y = 643.00(1.05)^x$

Step5: Predict initial value ($x=0$)

Substitute $x=0$ into the function: $y = 643.00(1.05)^0 = 643.00*1 = 643$

Answer:

$y = 643.00(1.05)^x$
$\$643$