Question

or
Find the equation for the least squares regression line of the data described below.
The owner of Grizzly Tea Shack is thinking about adding iced tea to the menu. She thinks she can do this with minimal effort by adding ice cubes to cups of hot tea. She decides to measure how changing the number of ice cubes in a glass of freshly brewed tea affects its cooling rate.
To begin, the owner varies the number of ice cubes, x, she puts in glasses of freshly brewed tea. She then checks the temperature (in Celsius), y, of each glass after 10 minutes.
Ice cubes Temperature after 10 minutes (in degrees Celsius)
1 18
2 5
4 10
7 12
7 10
Round your answers to the nearest thousandth.
y=
x+

Answer 1

To find the least squares regression line of the given data, we will first organize the data points and then apply the formulas for the slope (m) and the y-intercept (b) of the regression line.

### Data Points
We have the following pairs of \( (x, y) \):
- \( (1, 18) \)
- \( (2, 5) \)
- \( (4, 10) \)
- \( (7, 12) \)
- \( (7, 10) \)

### Step 1: Calculate the means
Calculate the means of \( x \) and \( y \).

\[
\text{Mean of } x (\bar{x}) = \frac{1 + 2 + 4 + 7 + 7}{5} = \frac{21}{5} = 4.2
\]

\[
\text{Mean of } y (\bar{y}) = \frac{18 + 5 + 10 + 12 + 10}{5} = \frac{55}{5} = 11
\]

### Step 2: Calculate the slope (m)
The slope \( m \) is calculated with the formula:

\[
m = \frac{N(\sum xy) - (\sum x)(\sum y)}{N(\sum x^2) - (\sum x)^2}
\]

#### Calculating each term:
- \( N = 5 \)
- \( \sum x = 1 + 2 + 4 + 7 + 7 = 21 \)
- \( \sum y = 18 + 5 + 10 + 12 + 10 = 55 \)
- \( \sum xy = (1)(18) + (2)(5) + (4)(10) + (7)(12) + (7)(10) = 18 + 10 + 40 + 84 + 70 = 232 \)
- \( \sum x^2 = 1^2 + 2^2 + 4^2 + 7^2 + 7^2 = 1 + 4 + 16 + 49 + 49 = 119 \)

#### Plugging into the slope formula:
\[
m = \frac{5(232) - (21)(55)}{5(119) - (21)^2}
\]
\[
m = \frac{1160 - 1155}{595 - 441} = \frac{5}{154}
\]
\[
m \approx 0.0325 \text{ (approx. to four decimal places)}
\]

### Step 3: Calculate the y-intercept (b)
The y-intercept \( b \) can be calculated using:

\[
b = \bar{y} - m\bar{x}
\]

\[
b = 11 - (0.0325)(4.2)
\]
\[
b = 11 - 0.1365 \approx 10.8635 \text{ (approx. to four decimal places)}
\]

### Step 4: Formulate the least squares regression line
Now that we have \( m \) and \( b \), our least squares regression line is in the form:

\[
y = mx + b
\]

Substituting the values we found:
\[
y = 0.0325x + 10.8635
\]

### Step 5: Rounding the coefficients
Now, rounding \( m \) and \( b \) to the nearest thousandth:

- \( m \approx 0.033 \)
- \( b \approx 10.864 \)

Thus, the final equation of the least squares regression line is:

\[
y = 0.033x + 10.864
\]

Feel free to ask if you need any more clarifications or additional assistance!