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 (4 points) Use the Desmos Graphing Calculator (right-click and open in new tab/window) to calculate the following:
r =
y = x +
If someone was 45 years old, what would the regression line predict for the strength of coffee they would prefer?
r =
y = x +
If someone was 45 years old, what would the regression line predict for the strength of coffee they would prefer?
Answers
GPT-4o mini
To determine the correlation coefficient (r) and the linear regression equation, we'll first need the data you've provided:
- Age of Subject (x): 17, 27, 33, 37, 40
- Strength of Coffee in grams (y): 48, 44, 49, 54, 57
Using this data, the next steps involve calculating the correlation coefficient and finding the regression line equation using a graphing calculator or statistical software.
1. **Calculate correlation coefficient (r)**: This quantifies the strength and direction of a linear relationship between the age of subjects and their preferred coffee strength.
2. **Determine linear regression line**: This is generally represented in the form \( y = mx + b \), where:
- \( m \) is the slope of the line,
- \( b \) is the y-intercept.
Using Desmos or any statistical calculator, you would input the data points and perform the regression analysis to get the values of \( r \), \( m \), and \( b \).
Assuming you've gone through the calculations:
### Hypothetical Results:
- Let's say you found \( r \approx 0.87 \) (indicating a strong positive correlation).
- Suppose the regression line equation is \( y = 0.5x + 40 \) (this is a made-up example for illustration).
### Prediction for a 45-year-old:
To find out the predicted strength of coffee for someone who is 45 years old, we would substitute \( x = 45 \) into the regression line equation.
Using our hypothetical equation \( y = 0.5x + 40 \):
\[ y = 0.5(45) + 40 \]
\[ y = 22.5 + 40 \]
\[ y = 62.5 \]
So, if we used the hypothetical equation, the prediction for the strength of coffee a 45-year-old would prefer would be approximately 62.5 grams.
### Actual Calculation:
You would need to perform this process using the actual data you provided in Desmos or another graphing calculator to get your actual values for \( r \) and the equation of the regression line. After that, substitute \( x = 45 \) into the found equation to get the prediction for the strength of coffee for someone who is 45 years old.
- Age of Subject (x): 17, 27, 33, 37, 40
- Strength of Coffee in grams (y): 48, 44, 49, 54, 57
Using this data, the next steps involve calculating the correlation coefficient and finding the regression line equation using a graphing calculator or statistical software.
1. **Calculate correlation coefficient (r)**: This quantifies the strength and direction of a linear relationship between the age of subjects and their preferred coffee strength.
2. **Determine linear regression line**: This is generally represented in the form \( y = mx + b \), where:
- \( m \) is the slope of the line,
- \( b \) is the y-intercept.
Using Desmos or any statistical calculator, you would input the data points and perform the regression analysis to get the values of \( r \), \( m \), and \( b \).
Assuming you've gone through the calculations:
### Hypothetical Results:
- Let's say you found \( r \approx 0.87 \) (indicating a strong positive correlation).
- Suppose the regression line equation is \( y = 0.5x + 40 \) (this is a made-up example for illustration).
### Prediction for a 45-year-old:
To find out the predicted strength of coffee for someone who is 45 years old, we would substitute \( x = 45 \) into the regression line equation.
Using our hypothetical equation \( y = 0.5x + 40 \):
\[ y = 0.5(45) + 40 \]
\[ y = 22.5 + 40 \]
\[ y = 62.5 \]
So, if we used the hypothetical equation, the prediction for the strength of coffee a 45-year-old would prefer would be approximately 62.5 grams.
### Actual Calculation:
You would need to perform this process using the actual data you provided in Desmos or another graphing calculator to get your actual values for \( r \) and the equation of the regression line. After that, substitute \( x = 45 \) into the found equation to get the prediction for the strength of coffee for someone who is 45 years old.