Question
Have you ever noticed that adults like the taste of some bitter foods that adolescents just cannot tolerate the taste of? Things like dark chocolate, grapefruit, coffee, etc. Ms. Wilson is interested in whether bitter flavors become more tolerable as people get older. She sets up a study in which subjects taste cups of coffee that are different strengths and picks their favorite one. Ms. Wilson records the age of each subject, x, and the strength of the coffee in grams, y, that the subject liked the best.
Age of Subject | Strength of Coffee in grams
17 | 48
27 | 44
33 | 49
37 | 54
40 | 57
Question: y = BLANKx + BLANK
Options for the first blank:
A. -0.728
B. 0.239
C. 0.788
D. 0.413
Options for the second blank:
A. 37.684
B. 41.338
C. 45.594
D. 47.331
Age of Subject | Strength of Coffee in grams
17 | 48
27 | 44
33 | 49
37 | 54
40 | 57
Question: y = BLANKx + BLANK
Options for the first blank:
A. -0.728
B. 0.239
C. 0.788
D. 0.413
Options for the second blank:
A. 37.684
B. 41.338
C. 45.594
D. 47.331
Answers
GPT-4o mini
To determine the linear regression equation \( y = mx + b \) for Ms. Wilson's data, we need to calculate the slope (\( m \)) and the y-intercept (\( b \)) using the provided data points.
The data points of age (x) and coffee strength (y) are:
- (17, 48)
- (27, 44)
- (33, 49)
- (37, 54)
- (40, 57)
1. **Calculate the means of x and y:**
\[
\text{Mean of } x = \frac{17 + 27 + 33 + 37 + 40}{5} = \frac{154}{5} = 30.8
\]
\[
\text{Mean of } y = \frac{48 + 44 + 49 + 54 + 57}{5} = \frac{252}{5} = 50.4
\]
2. **Calculate the sums needed for the slope \( m \):**
\[
S_{xx} = \sum (x_i - \text{mean } x)^2
\]
\[
S_{yy} = \sum (y_i - \text{mean } y)^2
\]
\[
S_{xy} = \sum (x_i - \text{mean } x)(y_i - \text{mean } y)
\]
Calculating these values:
- For \( x_i - \text{mean } x \):
- \( 17 - 30.8 = -13.8 \)
- \( 27 - 30.8 = -3.8 \)
- \( 33 - 30.8 = 2.2 \)
- \( 37 - 30.8 = 6.2 \)
- \( 40 - 30.8 = 9.2 \)
- For \( y_i - \text{mean } y \):
- \( 48 - 50.4 = -2.4 \)
- \( 44 - 50.4 = -6.4 \)
- \( 49 - 50.4 = -1.4 \)
- \( 54 - 50.4 = 3.6 \)
- \( 57 - 50.4 = 6.6 \)
Next, compute \( S_{xx} \), \( S_{yy} \), and \( S_{xy} \):
- \( S_{xx} = (-13.8)^2 + (-3.8)^2 + (2.2)^2 + (6.2)^2 + (9.2)^2 = 190.44 + 14.44 + 4.84 + 38.44 + 84.64 = 332.8 \)
- \( S_{yy} = (-2.4)^2 + (-6.4)^2 + (-1.4)^2 + (3.6)^2 + (6.6)^2 = 5.76 + 40.96 + 1.96 + 12.96 + 43.56 = 105.2 \)
- \( S_{xy} = (-13.8)(-2.4) + (-3.8)(-6.4) + (2.2)(-1.4) + (6.2)(3.6) + (9.2)(6.6) = 33.12 + 24.32 - 3.08 + 22.32 + 60.72 = 137.4 \)
3. **Calculate the slope \( m \)**:
\[
m = \frac{S_{xy}}{S_{xx}} = \frac{137.4}{332.8} \approx 0.413
\]
4. **Calculate the intercept \( b \)**:
Using \( b = \text{mean } y - m \cdot \text{mean } x \):
\[
b = 50.4 - 0.413 \cdot 30.8 \approx 50.4 - 12.7064 \approx 37.6936
\]
Thus, the regression equation is approximately:
\[
y = 0.413x + 37.694 \quad \text{(rounded to three decimal places)}
\]
The correct options to fill in the blanks:
- For the first blank (slope \( m \)): **D. 0.413**
- For the second blank (intercept \( b \)): **A. 37.684** (as it's the closest approximation)
So the final answer will be:
- \[ y = 0.413x + 37.684 \]
The data points of age (x) and coffee strength (y) are:
- (17, 48)
- (27, 44)
- (33, 49)
- (37, 54)
- (40, 57)
1. **Calculate the means of x and y:**
\[
\text{Mean of } x = \frac{17 + 27 + 33 + 37 + 40}{5} = \frac{154}{5} = 30.8
\]
\[
\text{Mean of } y = \frac{48 + 44 + 49 + 54 + 57}{5} = \frac{252}{5} = 50.4
\]
2. **Calculate the sums needed for the slope \( m \):**
\[
S_{xx} = \sum (x_i - \text{mean } x)^2
\]
\[
S_{yy} = \sum (y_i - \text{mean } y)^2
\]
\[
S_{xy} = \sum (x_i - \text{mean } x)(y_i - \text{mean } y)
\]
Calculating these values:
- For \( x_i - \text{mean } x \):
- \( 17 - 30.8 = -13.8 \)
- \( 27 - 30.8 = -3.8 \)
- \( 33 - 30.8 = 2.2 \)
- \( 37 - 30.8 = 6.2 \)
- \( 40 - 30.8 = 9.2 \)
- For \( y_i - \text{mean } y \):
- \( 48 - 50.4 = -2.4 \)
- \( 44 - 50.4 = -6.4 \)
- \( 49 - 50.4 = -1.4 \)
- \( 54 - 50.4 = 3.6 \)
- \( 57 - 50.4 = 6.6 \)
Next, compute \( S_{xx} \), \( S_{yy} \), and \( S_{xy} \):
- \( S_{xx} = (-13.8)^2 + (-3.8)^2 + (2.2)^2 + (6.2)^2 + (9.2)^2 = 190.44 + 14.44 + 4.84 + 38.44 + 84.64 = 332.8 \)
- \( S_{yy} = (-2.4)^2 + (-6.4)^2 + (-1.4)^2 + (3.6)^2 + (6.6)^2 = 5.76 + 40.96 + 1.96 + 12.96 + 43.56 = 105.2 \)
- \( S_{xy} = (-13.8)(-2.4) + (-3.8)(-6.4) + (2.2)(-1.4) + (6.2)(3.6) + (9.2)(6.6) = 33.12 + 24.32 - 3.08 + 22.32 + 60.72 = 137.4 \)
3. **Calculate the slope \( m \)**:
\[
m = \frac{S_{xy}}{S_{xx}} = \frac{137.4}{332.8} \approx 0.413
\]
4. **Calculate the intercept \( b \)**:
Using \( b = \text{mean } y - m \cdot \text{mean } x \):
\[
b = 50.4 - 0.413 \cdot 30.8 \approx 50.4 - 12.7064 \approx 37.6936
\]
Thus, the regression equation is approximately:
\[
y = 0.413x + 37.694 \quad \text{(rounded to three decimal places)}
\]
The correct options to fill in the blanks:
- For the first blank (slope \( m \)): **D. 0.413**
- For the second blank (intercept \( b \)): **A. 37.684** (as it's the closest approximation)
So the final answer will be:
- \[ y = 0.413x + 37.684 \]