Question

based on the data shown below, calculate the regression line (each value to two decimal places)
y = x +
x | y
3 | 10.58
4 | 12.64
5 | 14.9
6 | 14.86
7 | 14.62
8 | 17.48
9 | 18.24
10 | 20.6
11 | 22.26
12 | 22.32
13 | 25.48
14 | 28.04
15 | 28.1
16 | 28.66

Explanation:

Step 1: Calculate the necessary sums

First, we need to calculate \( n \) (number of data points), \( \sum x \), \( \sum y \), \( \sum xy \), and \( \sum x^2 \).

The data points are:
\( x: 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 \)
\( y: 10.58, 12.64, 14.9, 14.86, 14.62, 17.48, 18.24, 20.6, 22.26, 22.32, 25.48, 28.04, 28.1, 28.66 \)

\( n = 14 \)

\( \sum x = 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 \)
\[
\sum x = \frac{14(3 + 16)}{2} = 133
\]

\( \sum y = 10.58 + 12.64 + 14.9 + 14.86 + 14.62 + 17.48 + 18.24 + 20.6 + 22.26 + 22.32 + 25.48 + 28.04 + 28.1 + 28.66 \)
\[
\sum y = 263.88
\]

\( \sum xy = (3 \times 10.58) + (4 \times 12.64) + (5 \times 14.9) + (6 \times 14.86) + (7 \times 14.62) + (8 \times 17.48) + (9 \times 18.24) + (10 \times 20.6) + (11 \times 22.26) + (12 \times 22.32) + (13 \times 25.48) + (14 \times 28.04) + (15 \times 28.1) + (16 \times 28.66) \)
\[
\sum xy = 31.74 + 50.56 + 74.5 + 89.16 + 102.34 + 139.84 + 164.16 + 206 + 244.86 + 267.84 + 331.24 + 392.56 + 421.5 + 458.56 = 2784.86
\]

\( \sum x^2 = 3^2 + 4^2 + 5^2 + 6^2 + 7^2 + 8^2 + 9^2 + 10^2 + 11^2 + 12^2 + 13^2 + 14^2 + 15^2 + 16^2 \)
\[
\sum x^2 = 9 + 16 + 25 + 36 + 49 + 64 + 81 + 100 + 121 + 144 + 169 + 196 + 225 + 256 = 1491
\]

Step 2: Calculate the slope \( m \)

The formula for the slope \( m \) of the regression line is:
\[
m = \frac{n\sum xy - \sum x \sum y}{n\sum x^2 - (\sum x)^2}
\]

Substitute the values:
\[
m = \frac{14 \times 2784.86 - 133 \times 263.88}{14 \times 1491 - 133^2}
\]

First, calculate the numerator:
\( 14 \times 2784.86 = 38988.04 \)
\( 133 \times 263.88 = 35096.04 \)
Numerator: \( 38988.04 - 35096.04 = 3892 \)

Denominator:
\( 14 \times 1491 = 20874 \)
\( 133^2 = 17689 \)
Denominator: \( 20874 - 17689 = 3185 \)

So, \( m = \frac{3892}{3185} \approx 1.22 \) (rounded to two decimal places)

Step 3: Calculate the y-intercept \( b \)

The formula for the y-intercept \( b \) is:
\[
b = \frac{\sum y - m \sum x}{n}
\]

Substitute the values:
\[
b = \frac{263.88 - 1.22 \times 133}{14}
\]

Calculate \( 1.22 \times 133 = 162.26 \)
\( 263.88 - 162.26 = 101.62 \)
\( b = \frac{101.62}{14} \approx 7.26 \) (rounded to two decimal places)

Answer:

\( y = 1.22x + 7.26 \)