The table below shows data that has been collected from different fields on various farms in a certain valley. The table contains the grams of raspberries tested and the amount of their Vitamin C content in mg. Find a linear model that expresses Vitamin C content as a function of the weight of the raspberries.

\begin{tabular}{|r|r|}
\hline
grams & \begin{tabular}{c}
Vitamin C \\
content in mg
\end{tabular} \\
\hline
70 & 18.7 \\
\hline
80 & 23 \\
\hline
90 & 27.2 \\
\hline
100 & 32 \\
\hline
110 & 36.5 \\
\hline
120 & 40.6 \\
\hline
130 & 46 \\
\hline
\end{tabular}

A) Find the regression equation: [tex]y = \square x + \square[/tex]. Round your answers to 3 decimal places.

B) Answer the following question using your unrounded regression equation:

If we test 160 grams of raspberries, what is the expected Vitamin C content? [tex]\square \, \text{mg}[/tex] (round to the nearest tenth).



Answer :

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].