Question

the table shows the number of people participating in a charity walk over the past several years.
year 2008 2010 2016 2020 2023
number of participants 400 490 626 874 1152
calculate the exponential regression model for the data:
use your model to predict the number of participants in the year 2025

Explanation:

Step1: Define transformed variable $x$

Let $x$ = Year - 2008, so $x$ values are: $0, 2, 8, 12, 15$
Corresponding $y$ (participants): $400, 490, 626, 874, 1152$

Step2: Linearize exponential model

Exponential model: $y = ab^x$. Take natural log:
$\ln(y) = \ln(a) + x\ln(b)$
Let $Y = \ln(y)$, $A = \ln(a)$, $B = \ln(b)$, so $Y = A + Bx$

Step3: Calculate required sums

$\sum x = 0+2+8+12+15 = 37$
$\sum Y = \ln(400)+\ln(490)+\ln(626)+\ln(874)+\ln(1152)$
$\approx 5.9915 + 6.1944 + 6.4401 + 6.7727 + 7.0493 = 32.448$
$\sum x^2 = 0^2+2^2+8^2+12^2+15^2 = 0+4+64+144+225 = 437$
$\sum xY = (0\times5.9915)+(2\times6.1944)+(8\times6.4401)+(12\times6.7727)+(15\times7.0493)$
$\approx 0 + 12.3888 + 51.5208 + 81.2724 + 105.7395 = 250.9215$
$n = 5$

Step4: Solve for $B$ and $A$

$B = \frac{n\sum xY - \sum x\sum Y}{n\sum x^2 - (\sum x)^2}$
$= \frac{5\times250.9215 - 37\times32.448}{5\times437 - 37^2}$
$= \frac{1254.6075 - 1190.576}{2185 - 1369} = \frac{64.0315}{816} \approx 0.07859$

$A = \frac{\sum Y - B\sum x}{n}$
$= \frac{32.448 - 0.07859\times37}{5}$
$= \frac{32.448 - 2.9078}{5} = \frac{29.5402}{5} \approx 5.9080$

Step5: Find $a$ and $b$

$a = e^A = e^{5.9080} \approx 367.2$
$b = e^B = e^{0.07859} \approx 1.0818$
Exponential model: $y = 367.2(1.0818)^x$, where $x=\text{Year}-2008$

Step6: Predict 2025 participants

For 2025, $x = 2025-2008 = 17$
$y = 367.2(1.0818)^{17}$
$(1.0818)^{17} \approx e^{17\times0.07859} = e^{1.3360} \approx 3.802$
$y \approx 367.2\times3.802 \approx 1396$

Answer:

Exponential regression model: $\boldsymbol{y = 367.2(1.0818)^{x}}$ where $x = \text{Year} - 2008$
Predicted participants in 2025: $\boldsymbol{1396}$