Question

1. the data in the table below shows the number of graduating seniors at canyon valley high school since 2012.

year	graduates
2013	348
2014	356
2015	361
2016	375
2017	387

a) find the line of best fit:
b) estimate the number of graduating seniors in 2025.

Explanation:

Part (a)

Step 1: Assign Variables

Let \( x \) be the number of years since 2012 (so \( x = 0 \) for 2012, \( x = 1 \) for 2013, etc.), and \( y \) be the number of graduates. The data points are: \((0, 340)\), \((1, 348)\), \((2, 356)\), \((3, 361)\), \((4, 375)\), \((5, 387)\).

Step 2: Calculate Mean of \( x \) and \( y \)

Mean of \( x \) (\(\bar{x}\)): \(\frac{0 + 1 + 2 + 3 + 4 + 5}{6} = \frac{15}{6} = 2.5\)
Mean of \( y \) (\(\bar{y}\)): \(\frac{340 + 348 + 356 + 361 + 375 + 387}{6} = \frac{2167}{6} \approx 361.17\)

Step 3: Calculate Slope (\( m \))

Slope formula: \( m = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2} \)
Calculate numerator:
\((0 - 2.5)(340 - 361.17) + (1 - 2.5)(348 - 361.17) + (2 - 2.5)(356 - 361.17) + (3 - 2.5)(361 - 361.17) + (4 - 2.5)(375 - 361.17) + (5 - 2.5)(387 - 361.17)\)
\(= (-2.5)(-21.17) + (-1.5)(-13.17) + (-0.5)(-5.17) + (0.5)(-0.17) + (1.5)(13.83) + (2.5)(25.83)\)
\(= 52.925 + 19.755 + 2.585 - 0.085 + 20.745 + 64.575 = 160.5\)

Denominator:
\((0 - 2.5)^2 + (1 - 2.5)^2 + (2 - 2.5)^2 + (3 - 2.5)^2 + (4 - 2.5)^2 + (5 - 2.5)^2\)
\(= 6.25 + 2.25 + 0.25 + 0.25 + 2.25 + 6.25 = 17.5\)

Slope \( m = \frac{160.5}{17.5} \approx 9.17\)

Step 4: Calculate Y-Intercept (\( b \))

Using \( \bar{y} = m\bar{x} + b \):
\( 361.17 = 9.17(2.5) + b \)
\( 361.17 = 22.925 + b \)
\( b = 361.17 - 22.925 \approx 338.245 \)

Step 5: Line of Best Fit

The equation is \( y = 9.17x + 338.25 \) (or rounded for simplicity, \( y \approx 9.17x + 338.25 \)).

Part (b)

Step 1: Determine \( x \) for 2025

2025 - 2012 = 13, so \( x = 13 \).

Step 2: Substitute \( x = 13 \) into Line of Best Fit

Using \( y = 9.17x + 338.25 \):
\( y = 9.17(13) + 338.25 \)
\( y = 119.21 + 338.25 = 457.46 \)

Final Answers

a) Line of best fit: \( \boldsymbol{y \approx 9.17x + 338.25} \) (where \( x \) = years since 2012)
b) Estimated graduates in 2025: \( \boldsymbol{\approx 457} \) (or 457.46, rounded appropriately)

Answer:

Step 1: Determine \( x \) for 2025

2025 - 2012 = 13, so \( x = 13 \).

Step 2: Substitute \( x = 13 \) into Line of Best Fit

Using \( y = 9.17x + 338.25 \):
\( y = 9.17(13) + 338.25 \)
\( y = 119.21 + 338.25 = 457.46 \)

Final Answers

a) Line of best fit: \( \boldsymbol{y \approx 9.17x + 338.25} \) (where \( x \) = years since 2012)
b) Estimated graduates in 2025: \( \boldsymbol{\approx 457} \) (or 457.46, rounded appropriately)