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]
