Question
Find the equation for the least squares regression line of the data described below.
Dominic wants to ensure that the new board game he's designing is accessible to a wide age range. To test this, he had several children of various ages play the board game against each other. Each child played the game the same number of times.
For each child, Dominic recorded his or her age, x, and the number of games that child had won, y.
Age Games won
8 3
9 4
10 4
11 8
14 5
Round your answers to the nearest thousandth.
y = ___x + _____
Dominic wants to ensure that the new board game he's designing is accessible to a wide age range. To test this, he had several children of various ages play the board game against each other. Each child played the game the same number of times.
For each child, Dominic recorded his or her age, x, and the number of games that child had won, y.
Age Games won
8 3
9 4
10 4
11 8
14 5
Round your answers to the nearest thousandth.
y = ___x + _____
Answers
GPT-4o mini
To find the equation of the least squares regression line, we first need to calculate the necessary statistics based on the provided data.
Given the data:
- (8, 3)
- (9, 4)
- (10, 4)
- (11, 8)
- (14, 5)
**Step 1: Calculate the sums needed for the regression coefficients.**
Let \( n \) be the number of data points, \( \sum x \) be the sum of the x values (ages), \( \sum y \) be the sum of the y values (games won), \( \sum xy \) be the sum of the product of x and y, and \( \sum x^2 \) be the sum of the squares of x.
Calculate the individual sums:
- \( n = 5 \)
- \( \sum x = 8 + 9 + 10 + 11 + 14 = 52 \)
- \( \sum y = 3 + 4 + 4 + 8 + 5 = 24 \)
- \( \sum xy = (8 \cdot 3) + (9 \cdot 4) + (10 \cdot 4) + (11 \cdot 8) + (14 \cdot 5) = 24 + 36 + 40 + 88 + 70 = 258 \)
- \( \sum x^2 = 8^2 + 9^2 + 10^2 + 11^2 + 14^2 = 64 + 81 + 100 + 121 + 196 = 562 \)
**Step 2: Compute the slope \( b \) and y-intercept \( a \) for the regression line.**
The formulas for the slope \( b \) and y-intercept \( a \) are as follows:
\[
b = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}
\]
\[
a = \frac{\sum y - b(\sum x)}{n}
\]
**Substituting in the values:**
- \( n = 5 \)
- \( \sum x = 52 \)
- \( \sum y = 24 \)
- \( \sum xy = 258 \)
- \( \sum x^2 = 562 \)
Calculating \( b \):
\[
b = \frac{5(258) - (52)(24)}{5(562) - (52)^2}
\]
Calculating \( 5(258) = 1290 \)
Calculating \( (52)(24) = 1248 \)
Calculating the numerator:
\[
1290 - 1248 = 42
\]
Calculating \( 5(562) = 2810 \)
Calculating \( (52)^2 = 2704 \)
Calculating the denominator:
\[
2810 - 2704 = 106
\]
Therefore,
\[
b = \frac{42}{106} \approx 0.396226 \text{ (rounded to 5 decimal places)}
\]
Now let's compute \( a \):
\[
a = \frac{24 - b(52)}{5}
\]
Calculating \( b(52) \):
\[
0.396226 \cdot 52 \approx 20.570
\]
Calculating \( 24 - 20.570 = 3.43 \)
Now substitute into the formula for \( a \):
\[
a = \frac{3.43}{5} \approx 0.686 \text{ (rounded to 3 decimal places)}
\]
**Final Equation:**
Putting it all together, the equation for the least squares regression line is:
\[
y = 0.396x + 0.686
\]
Thus, the answer is:
\(y = 0.396x + 0.686\) (rounded to the nearest thousandth).
Given the data:
- (8, 3)
- (9, 4)
- (10, 4)
- (11, 8)
- (14, 5)
**Step 1: Calculate the sums needed for the regression coefficients.**
Let \( n \) be the number of data points, \( \sum x \) be the sum of the x values (ages), \( \sum y \) be the sum of the y values (games won), \( \sum xy \) be the sum of the product of x and y, and \( \sum x^2 \) be the sum of the squares of x.
Calculate the individual sums:
- \( n = 5 \)
- \( \sum x = 8 + 9 + 10 + 11 + 14 = 52 \)
- \( \sum y = 3 + 4 + 4 + 8 + 5 = 24 \)
- \( \sum xy = (8 \cdot 3) + (9 \cdot 4) + (10 \cdot 4) + (11 \cdot 8) + (14 \cdot 5) = 24 + 36 + 40 + 88 + 70 = 258 \)
- \( \sum x^2 = 8^2 + 9^2 + 10^2 + 11^2 + 14^2 = 64 + 81 + 100 + 121 + 196 = 562 \)
**Step 2: Compute the slope \( b \) and y-intercept \( a \) for the regression line.**
The formulas for the slope \( b \) and y-intercept \( a \) are as follows:
\[
b = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}
\]
\[
a = \frac{\sum y - b(\sum x)}{n}
\]
**Substituting in the values:**
- \( n = 5 \)
- \( \sum x = 52 \)
- \( \sum y = 24 \)
- \( \sum xy = 258 \)
- \( \sum x^2 = 562 \)
Calculating \( b \):
\[
b = \frac{5(258) - (52)(24)}{5(562) - (52)^2}
\]
Calculating \( 5(258) = 1290 \)
Calculating \( (52)(24) = 1248 \)
Calculating the numerator:
\[
1290 - 1248 = 42
\]
Calculating \( 5(562) = 2810 \)
Calculating \( (52)^2 = 2704 \)
Calculating the denominator:
\[
2810 - 2704 = 106
\]
Therefore,
\[
b = \frac{42}{106} \approx 0.396226 \text{ (rounded to 5 decimal places)}
\]
Now let's compute \( a \):
\[
a = \frac{24 - b(52)}{5}
\]
Calculating \( b(52) \):
\[
0.396226 \cdot 52 \approx 20.570
\]
Calculating \( 24 - 20.570 = 3.43 \)
Now substitute into the formula for \( a \):
\[
a = \frac{3.43}{5} \approx 0.686 \text{ (rounded to 3 decimal places)}
\]
**Final Equation:**
Putting it all together, the equation for the least squares regression line is:
\[
y = 0.396x + 0.686
\]
Thus, the answer is:
\(y = 0.396x + 0.686\) (rounded to the nearest thousandth).