Question

find the equation for the least squares regression line of the data described below. researchers at a cognitive psychology lab are studying how children develop their color vocabulary. in a recent study, the researchers showed children of various ages different colored objects and asked them to name the colors. they recorded each childs age (in months), x, and the number of colors the child could name, y. age (in months) colors 28 5 33 5 34 4 36 5 45 9 round your answers to the nearest thousandth. y = x +

Explanation:

Step1: Calculate necessary sums

First, we list the data points: \((28, 5)\), \((33, 5)\), \((34, 4)\), \((36, 5)\), \((45, 9)\)

Number of data points \(n = 5\)

Calculate \(\sum x\), \(\sum y\), \(\sum xy\), \(\sum x^2\)

\(\sum x = 28 + 33 + 34 + 36 + 45 = 176\)

\(\sum y = 5 + 5 + 4 + 5 + 9 = 28\)

\(\sum xy = (28\times5) + (33\times5) + (34\times4) + (36\times5) + (45\times9)\)
\(= 140 + 165 + 136 + 180 + 405 = 1026\)

\(\sum x^2 = 28^2 + 33^2 + 34^2 + 36^2 + 45^2\)
\(= 784 + 1089 + 1156 + 1296 + 2025 = 6350\)

Step2: Calculate 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:
\(m = \frac{5\times1026 - 176\times28}{5\times6350 - 176^2}\)
First, calculate numerator: \(5\times1026 = 5130\), \(176\times28 = 4928\), so numerator \(= 5130 - 4928 = 202\)

Denominator: \(5\times6350 = 31750\), \(176^2 = 30976\), so denominator \(= 31750 - 30976 = 774\)

Then \(m = \frac{202}{774} \approx 0.261\)

Step3: Calculate y-intercept \(b\)

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

Substitute the values:
\(b = \frac{28 - 0.261\times176}{5}\)
First, calculate \(0.261\times176 \approx 45.936\)

Then \(28 - 45.936 = -17.936\)

\(b = \frac{-17.936}{5} \approx -3.587\)

Answer:

\(y = 0.261x - 3.587\)