Question
or
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 child's age (in months), x, and the number of colors the child could name, y.
Age (in months) Colors
24 3
35 5
39 9
40 6
45 9
Round your answers to the nearest thousandth.
y=
x+
Answers
Answered by
GPT-4o mini
To create the least squares regression line \(y = mx + b\), we need to calculate the slope \(m\) and the y-intercept \(b\). Here's how to do that step-by-step:
### Step 1: Collect the data points
We have the following data points:
- (24, 3)
- (35, 5)
- (39, 9)
- (40, 6)
- (45, 9)
### Step 2: Calculate the required sums
We need to calculate the following:
- \(n\) = number of data points
- \(\Sigma x\) = sum of \(x\) values
- \(\Sigma y\) = sum of \(y\) values
- \(\Sigma xy\) = sum of the product of each \(x\) and \(y\)
- \(\Sigma x^2\) = sum of \(x\) squared
Using the data points mentioned:
- \(\Sigma x = 24 + 35 + 39 + 40 + 45 = 183\)
- \(\Sigma y = 3 + 5 + 9 + 6 + 9 = 32\)
- \(\Sigma xy = (24*3) + (35*5) + (39*9) + (40*6) + (45*9) = 72 + 175 + 351 + 240 + 405 = 1243\)
- \(\Sigma x^2 = (24^2) + (35^2) + (39^2) + (40^2) + (45^2) = 576 + 1225 + 1521 + 1600 + 2025 = 5947\)
Now, calculate \(n\):
- \(n = 5\)
### Step 3: Calculate the slope \(m\) and y-intercept \(b\)
The formulas for the slope \(m\) and y-intercept \(b\) of the least squares regression line are:
\[
m = \frac{n(\Sigma xy) - (\Sigma x)(\Sigma y)}{n(\Sigma x^2) - (\Sigma x)^2}
\]
\[
b = \frac{(\Sigma y)(\Sigma x^2) - (\Sigma x)(\Sigma xy)}{n(\Sigma x^2) - (\Sigma x)^2}
\]
Substituting the values we calculated:
\[
m = \frac{5(1243) - (183)(32)}{5(5947) - (183)^2}
\]
Calculate each component:
1. \(5(1243) = 6215\)
2. \(183(32) = 5856\)
3. \(5(5947) = 29735\)
4. \(183^2 = 33489\)
Now put these into the slope formula:
\[
m = \frac{6215 - 5856}{29735 - 33489} = \frac{359}{-3764} \approx -0.0952
\]
Now for \(b\):
\[
b = \frac{(32)(5947) - (183)(1243)}{5(5947) - (183)^2}
\]
Calculate:
1. \(32(5947) = 190304\)
2. \(183(1243) = 227289\)
Now substitute these into the intercept formula:
\[
b = \frac{190304 - 227289}{29735 - 33489} = \frac{-36985}{-3764} \approx 9.812
\]
### Final Equation
Rounding \(m\) and \(b\) to the nearest thousandth, we have:
\[
y \approx -0.095 x + 9.812
\]
So the equation of the least squares regression line is:
\[
y = -0.095x + 9.812
\]
### Step 1: Collect the data points
We have the following data points:
- (24, 3)
- (35, 5)
- (39, 9)
- (40, 6)
- (45, 9)
### Step 2: Calculate the required sums
We need to calculate the following:
- \(n\) = number of data points
- \(\Sigma x\) = sum of \(x\) values
- \(\Sigma y\) = sum of \(y\) values
- \(\Sigma xy\) = sum of the product of each \(x\) and \(y\)
- \(\Sigma x^2\) = sum of \(x\) squared
Using the data points mentioned:
- \(\Sigma x = 24 + 35 + 39 + 40 + 45 = 183\)
- \(\Sigma y = 3 + 5 + 9 + 6 + 9 = 32\)
- \(\Sigma xy = (24*3) + (35*5) + (39*9) + (40*6) + (45*9) = 72 + 175 + 351 + 240 + 405 = 1243\)
- \(\Sigma x^2 = (24^2) + (35^2) + (39^2) + (40^2) + (45^2) = 576 + 1225 + 1521 + 1600 + 2025 = 5947\)
Now, calculate \(n\):
- \(n = 5\)
### Step 3: Calculate the slope \(m\) and y-intercept \(b\)
The formulas for the slope \(m\) and y-intercept \(b\) of the least squares regression line are:
\[
m = \frac{n(\Sigma xy) - (\Sigma x)(\Sigma y)}{n(\Sigma x^2) - (\Sigma x)^2}
\]
\[
b = \frac{(\Sigma y)(\Sigma x^2) - (\Sigma x)(\Sigma xy)}{n(\Sigma x^2) - (\Sigma x)^2}
\]
Substituting the values we calculated:
\[
m = \frac{5(1243) - (183)(32)}{5(5947) - (183)^2}
\]
Calculate each component:
1. \(5(1243) = 6215\)
2. \(183(32) = 5856\)
3. \(5(5947) = 29735\)
4. \(183^2 = 33489\)
Now put these into the slope formula:
\[
m = \frac{6215 - 5856}{29735 - 33489} = \frac{359}{-3764} \approx -0.0952
\]
Now for \(b\):
\[
b = \frac{(32)(5947) - (183)(1243)}{5(5947) - (183)^2}
\]
Calculate:
1. \(32(5947) = 190304\)
2. \(183(1243) = 227289\)
Now substitute these into the intercept formula:
\[
b = \frac{190304 - 227289}{29735 - 33489} = \frac{-36985}{-3764} \approx 9.812
\]
### Final Equation
Rounding \(m\) and \(b\) to the nearest thousandth, we have:
\[
y \approx -0.095 x + 9.812
\]
So the equation of the least squares regression line is:
\[
y = -0.095x + 9.812
\]
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