Question
Find the equation for the least squares regression line of the data described below.
Doug hangs his laundry outside every Saturday and has noticed that his clothes dry faster on some days than others. He is curious to know how the daily temperature influences the time it takes for his clothes to dry.
Over the last several Saturdays, he recorded the temperature (in Celsius), x, and the time it had taken for his clothes to dry (in hours), y.
Temperature Drying time
17 19
22 13
28 16
31 13
33 12
Round your answers to the nearest thousandth.
y=
x+
Answers
GPT-4o mini
To find the least squares regression line for the given data, we first need to calculate several key components: the means of \( x \) and \( y \), the slope \( m \), and the y-intercept \( b \) of the regression line \( y = mx + b \).
Given data points:
\[
\begin{align*}
(17, 19) \\
(22, 13) \\
(28, 16) \\
(31, 13) \\
(33, 12) \\
\end{align*}
\]
### Step 1: Calculate the means of \( x \) and \( y \)
\[
\text{Mean of } x = \frac{17 + 22 + 28 + 31 + 33}{5} = \frac{131}{5} = 26.2
\]
\[
\text{Mean of } y = \frac{19 + 13 + 16 + 13 + 12}{5} = \frac{73}{5} = 14.6
\]
### Step 2: Calculate the slope \( m \)
The slope \( m \) can be calculated using the formula:
\[
m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}
\]
Where:
- \( n \) is the number of points
- \( \sum xy \) is the sum of the products of \( x \) and \( y \)
- \( \sum x \) is the sum of the \( x \) values
- \( \sum y \) is the sum of the \( y \) values
- \( \sum x^2 \) is the sum of the squares of \( x \)
Calculating \( \sum x \), \( \sum y \), \( \sum xy \), and \( \sum x^2 \):
\[
\sum x = 17 + 22 + 28 + 31 + 33 = 131
\]
\[
\sum y = 19 + 13 + 16 + 13 + 12 = 73
\]
\[
\sum xy = (17 \cdot 19) + (22 \cdot 13) + (28 \cdot 16) + (31 \cdot 13) + (33 \cdot 12)
\]
Calculating each term:
\[
17 \cdot 19 = 323,
\quad 22 \cdot 13 = 286,
\quad 28 \cdot 16 = 448,
\quad 31 \cdot 13 = 403,
\quad 33 \cdot 12 = 396
\]
Summing these products:
\[
\sum xy = 323 + 286 + 448 + 403 + 396 = 1856
\]
Now calculating \( \sum x^2 \):
\[
\sum x^2 = 17^2 + 22^2 + 28^2 + 31^2 + 33^2 = 289 + 484 + 784 + 961 + 1089 = 2587
\]
Now plug in values:
- \( n = 5 \)
- \( \sum x = 131 \)
- \( \sum y = 73 \)
- \( \sum xy = 1856 \)
- \( \sum x^2 = 2587 \)
Calculating \( m \):
\[
m = \frac{5(1856) - (131)(73)}{5(2587) - (131)^2} = \frac{9280 - 9583}{12935 - 17161} = \frac{-303}{-4216} = 0.0718 \text{ (approximately)}
\]
### Step 3: Calculate the y-intercept \( b \)
The y-intercept \( b \) can be calculated using the formula:
\[
b = \bar{y} - m \bar{x}
\]
Using calculated values:
\[
b = 14.6 - (0.0718)(26.2) = 14.6 - 1.883 \approx 12.717 \text{ (approximately)}
\]
### Final Equation
Now we can write the regression equation:
\[
y \approx 0.072x + 12.717
\]
Rounding to the nearest thousandth:
\[
y = 0.072x + 12.717
\]
So the final answer is:
\[
\boxed{y = 0.072x + 12.717}
\]
Given data points:
\[
\begin{align*}
(17, 19) \\
(22, 13) \\
(28, 16) \\
(31, 13) \\
(33, 12) \\
\end{align*}
\]
### Step 1: Calculate the means of \( x \) and \( y \)
\[
\text{Mean of } x = \frac{17 + 22 + 28 + 31 + 33}{5} = \frac{131}{5} = 26.2
\]
\[
\text{Mean of } y = \frac{19 + 13 + 16 + 13 + 12}{5} = \frac{73}{5} = 14.6
\]
### Step 2: Calculate the slope \( m \)
The slope \( m \) can be calculated using the formula:
\[
m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}
\]
Where:
- \( n \) is the number of points
- \( \sum xy \) is the sum of the products of \( x \) and \( y \)
- \( \sum x \) is the sum of the \( x \) values
- \( \sum y \) is the sum of the \( y \) values
- \( \sum x^2 \) is the sum of the squares of \( x \)
Calculating \( \sum x \), \( \sum y \), \( \sum xy \), and \( \sum x^2 \):
\[
\sum x = 17 + 22 + 28 + 31 + 33 = 131
\]
\[
\sum y = 19 + 13 + 16 + 13 + 12 = 73
\]
\[
\sum xy = (17 \cdot 19) + (22 \cdot 13) + (28 \cdot 16) + (31 \cdot 13) + (33 \cdot 12)
\]
Calculating each term:
\[
17 \cdot 19 = 323,
\quad 22 \cdot 13 = 286,
\quad 28 \cdot 16 = 448,
\quad 31 \cdot 13 = 403,
\quad 33 \cdot 12 = 396
\]
Summing these products:
\[
\sum xy = 323 + 286 + 448 + 403 + 396 = 1856
\]
Now calculating \( \sum x^2 \):
\[
\sum x^2 = 17^2 + 22^2 + 28^2 + 31^2 + 33^2 = 289 + 484 + 784 + 961 + 1089 = 2587
\]
Now plug in values:
- \( n = 5 \)
- \( \sum x = 131 \)
- \( \sum y = 73 \)
- \( \sum xy = 1856 \)
- \( \sum x^2 = 2587 \)
Calculating \( m \):
\[
m = \frac{5(1856) - (131)(73)}{5(2587) - (131)^2} = \frac{9280 - 9583}{12935 - 17161} = \frac{-303}{-4216} = 0.0718 \text{ (approximately)}
\]
### Step 3: Calculate the y-intercept \( b \)
The y-intercept \( b \) can be calculated using the formula:
\[
b = \bar{y} - m \bar{x}
\]
Using calculated values:
\[
b = 14.6 - (0.0718)(26.2) = 14.6 - 1.883 \approx 12.717 \text{ (approximately)}
\]
### Final Equation
Now we can write the regression equation:
\[
y \approx 0.072x + 12.717
\]
Rounding to the nearest thousandth:
\[
y = 0.072x + 12.717
\]
So the final answer is:
\[
\boxed{y = 0.072x + 12.717}
\]