3 the accompanying table shows the enrollment of a preschool from 1980 through 2000. write a linear regression equation to model the data in the table.
| year (x) | enrollment (y) |
| ---- | ---- |
| 1980 | 14 |
| 1985 | 20 |
| 1990 | 22 |
| 1995 | 28 |
| 2000 | 37 |

Question

Accepted Answer

# Explanation:
## Step1: Define variables and data points
Let $x$ be the year. We can re - scale $x$ so that $x = 0$ represents 1980. Then the data points are $(0,14)$, $(5,20)$, $(10,22)$, $(15,28)$, $(20,37)$.
## Step2: Calculate the means of $x$ and $y$
$\bar{x}=\frac{0 + 5+10+15+20}{5}=10$
$\bar{y}=\frac{14 + 20+22+28+37}{5}=\frac{121}{5}=24.2$
## Step3: Calculate the slope $m$
$$
\begin{align*}
m&=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})(y_i - \bar{y})}{\sum_{i=1}^{n}(x_i-\bar{x})^2}\
\sum_{i = 1}^{5}(x_i-\bar{x})(y_i - \bar{y})&=(0 - 10)(14 - 24.2)+(5 - 10)(20 - 24.2)+(10 - 10)(22 - 24.2)+(15 - 10)(28 - 24.2)+(20 - 10)(37 - 24.2)\
&=(- 10)\times(-10.2)+(-5)\times(-4.2)+0\times(-2.2)+5\times3.8+10\times12.8\
&=102 + 21+0 + 19+128\
&=270\
\sum_{i=1}^{5}(x_i-\bar{x})^2&=(0 - 10)^2+(5 - 10)^2+(10 - 10)^2+(15 - 10)^2+(20 - 10)^2\
&=100+25 + 0+25+100\
&=250\
m&=\frac{270}{250}=1.08
\end{align*}
$$
## Step4: Calculate the y - intercept $b$
Using the formula $y=mx + b$ and substituting $x=\bar{x}$, $y=\bar{y}$ and $m = 1.08$
$24.2=1.08\times10 + b$
$b=24.2-10.8 = 13.4$
## Step5: Write the linear regression equation
The linear regression equation is $y = 1.08x+13.4$, where $x$ is the number of years since 1980.

# Answer:
$y = 1.08x+13.4$ (where $x$ is the number of years since 1980)

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QUESTION IMAGE

Question

Explanation:

Step1: Define variables and data points

Step2: Calculate the means of \(x\) and \(y\)

Step3: Calculate the slope \(m\)

Step4: Calculate the y - intercept \(b\)

Step5: Write the linear regression equation

Answer: