To find a linear model that expresses Vitamin C content as a function of the weight of the Raspberries, we need to perform linear regression on the given data.
### A) Regression Equation
The given data points are:
| Grams (x) | Vitamin C (y) |
|-----------|---------------|
| 70 | 18.7 |
| 80 | 23.0 |
| 90 | 27.2 |
| 100 | 32.0 |
| 110 | 36.5 |
| 120 | 40.6 |
| 130 | 46.0 |
In linear regression, we aim to fit a line [tex]\( y = mx + b \)[/tex] through these points where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
After performing the linear regression calculations:
- The slope [tex]\( m \)[/tex] is calculated to be approximately 0.451.
- The y-intercept [tex]\( b \)[/tex] is calculated to be approximately -13.143.
So, the regression equation can be written as:
[tex]\[ y = 0.451x - 13.143 \][/tex]
### B) Expected Vitamin C Content for 160 Grams
Using the regression equation we derived earlier to predict the Vitamin C content for 160 grams of raspberries:
[tex]\[ y = 0.451 \times 160 - 13.143 \][/tex]
First, perform the multiplication and then the addition/subtraction:
[tex]\[ y = 72.16 - 13.143 \][/tex]
[tex]\[ y = 59.1 \][/tex]
Therefore, the expected Vitamin C content for 160 grams of raspberries is:
[tex]\[ 59.1 \, \text{mg} \][/tex]
To summarize:
1. The regression equation is [tex]\( y = 0.451x - 13.143 \)[/tex].
2. The expected Vitamin C content for 160 grams of raspberries is approximately [tex]\( 59.1 \, \text{mg} \)[/tex].
