Math Snap

STEP 1

1. The independent variable $x$ represents the year.
2. The dependent variable $y$ represents the tuition and fees in dollars.
3. We will use the least-squares method to find the regression line.
4. The years will be transformed into a form suitable for regression analysis, typically by setting the first year as $x = 0$.

STEP 2

1. Transform the year data for regression analysis.
2. Calculate the necessary sums for the least-squares formulas.
3. Compute the slope $m$ of the regression line.
4. Compute the y-intercept $b$ of the regression line.
5. Write the equation of the regression line.

STEP 3

Transform the year data by setting 1985 as $x = 0$. The transformed years are:
\begin{align} 1985 & \rightarrow 0 \\ 1990 & \rightarrow 5 \\ 1995 & \rightarrow 10 \\ 2000 & \rightarrow 15 \\ 2008 & \rightarrow 23 \\ \end{align}

STEP 4

Calculate the necessary sums:
\begin{align} \sum x &= 0 + 5 + 10 + 15 + 23 = 53 \\ \sum y &= 5328 + 9369 + 12336 + 16154 + 25111 = 68298 \\ \sum xy &= (0 \times 5328) + (5 \times 9369) + (10 \times 12336) + (15 \times 16154) + (23 \times 25111) = 100,7853 \\ \sum x^2 &= 0^2 + 5^2 + 10^2 + 15^2 + 23^2 = 899 \\ \end{align}

STEP 5

Compute the slope $m$ using the formula:
$$m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}$$ where $n = 5$.
$$m = \frac{5(1007853) - (53)(68298)}{5(899) - (53)^2} = \frac{5039265 - 3629794}{4495 - 2809} = \frac{1409471}{1686} \approx 835.993$$

STEP 6

Compute the y-intercept $b$ using the formula:
$$b = \frac{\sum y - m(\sum x)}{n}$$ $$b = \frac{68298 - 835.993 \times 53}{5} = \frac{68298 - 44207.629}{5} \approx 4818.874$$ Round $b$ to the nearest integer: $b \approx 4819$.

Write the equation of the regression line:
$$y \approx 835.993x + 4819$$ The equation of the least-squares regression line is:
$$y \approx 835.993x + 4819$$