The table below shows data 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 best-fitting 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 50 & 9.3 \\
\hline 60 & 13.7 \\
\hline 70 & 18.5 \\
\hline 80 & 23.2 \\
\hline 90 & 27 \\
\hline 100 & 32.6 \\
\hline 110 & 37.2 \\
\hline
\end{tabular}

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

B) If we test 140 grams of raspberries, what is the expected Vitamin C content? (Round to one decimal place as needed) [tex][tex]$\square$[/tex] $mg$[/tex]



Answer :

To address the problem, we need to find the best-fitting linear model for the given data that correlates the grams of raspberries tested ([tex]$x$[/tex]) to their Vitamin C content ([tex]$y$[/tex]) in mg.

Firstly, using linear regression, we find the best-fitting line's slope and intercept, denoted by coefficients [tex]\(m\)[/tex] and [tex]\(b\)[/tex] in the equation of the form:

[tex]\[ y = mx + b \][/tex]

Based on the computations:

A) The regression equation:
[tex]\[ y = 0.464x - 14.071 \][/tex]
where the slope [tex]\(m\)[/tex] is [tex]\(0.464\)[/tex] and the intercept [tex]\(b\)[/tex] is [tex]\(-14.071\)[/tex]. These values are rounded to 3 decimal places as required.

Now, let's use this equation to predict the Vitamin C content for 140 grams of raspberries.

Substitute [tex]\(x = 140\)[/tex] into the regression equation:

[tex]\[ y = 0.464(140) - 14.071 \][/tex]

Calculating the result:

[tex]\[ y = 64.96 - 14.071 = 50.889 \][/tex]

Rounding this to one decimal place, the expected Vitamin C content:

[tex]\[ 50.9 \text{ mg} \][/tex]

Thus, the expected Vitamin C content for 140 grams of raspberries is:

[tex]\[ 50.9 \text{ mg} \][/tex]

In summary:
1. The regression equation is:
[tex]\[ y = 0.464x - 14.071 \][/tex]
2. For 140 grams of raspberries, the expected Vitamin C content is:
[tex]\[ 50.9 \text{ mg} \][/tex]