Question

find the equation for the least squares regression line of the data described below.
ronald is writing an article for an online music magazine on why some bands last longer than others. while researching the article, he compiled information about how long various bands had lasted without changing their lineups.
one comparison he looked at was between the number of members in each band, x, and the number of years each band had lasted, y.
members | years
3 | 19
4 | 13
5 | 15
7 | 9
9 | 17
round your answers to the nearest thousandth.
y = \boxed{}x + \boxed{}

Explanation:

Step1: Calculate necessary sums

First, we list the data points: \((3, 19)\), \((4, 13)\), \((5, 15)\), \((7, 9)\), \((9, 17)\)

Calculate \(n = 5\) (number of data points).

Calculate \(\sum x\): \(3 + 4 + 5 + 7 + 9 = 28\)

Calculate \(\sum y\): \(19 + 13 + 15 + 9 + 17 = 73\)

Calculate \(\sum xy\): \(3\times19 + 4\times13 + 5\times15 + 7\times9 + 9\times17 = 57 + 52 + 75 + 63 + 153 = 390\)

Calculate \(\sum x^2\): \(3^2 + 4^2 + 5^2 + 7^2 + 9^2 = 9 + 16 + 25 + 49 + 81 = 180\)

Step2: Calculate the slope \(m\)

The formula for the slope \(m\) of the least squares regression line is:
\[
m = \frac{n\sum xy - \sum x \sum y}{n\sum x^2 - (\sum x)^2}
\]
Substitute the values:
\[
m = \frac{5\times390 - 28\times73}{5\times180 - 28^2} = \frac{1950 - 2044}{900 - 784} = \frac{-94}{116} \approx -0.810
\]

Step3: Calculate the y-intercept \(b\)

The formula for the y-intercept \(b\) is:
\[
b = \frac{\sum y - m\sum x}{n}
\]
Substitute the values:
\[
b = \frac{73 - (-0.810)\times28}{5} = \frac{73 + 22.68}{5} = \frac{95.68}{5} \approx 19.136
\]

Answer:

\(y = -0.810x + 19.136\)