Question

write the equation of the linear model.
$hat{y}=-0.418x + 7.93$
if you pick up a cereal and see that it has 3 grams of sugar, what is the predicted fiber content?
what is the residual for the cereal with 9 grams of sugar and 6 grams of fiber?
calculate the value of $r^{2}$ and interpret this value in context.
we know that the further you get from the equator, the colder the climate becomes. for the follow cities, we have assembled their latitude (angular distance from the equator, measured in degrees) and their average temperature in december of 2021. the data and the scatter - plot are given below.
city,latitude ($^{circ}$n),dec average temp ($^{circ}$f)
albany, ny,42.6,22.2
anchorage, ak,61.2,15.8
atlanta, ga,33.7,42.7
austin, tx,30.3,50.2
bismarck, nd,46.8,10.2
boise, id,43.6,30.2
boston, ma,42.4,29.3
charleston, sc,32.8,47.9
chicago, il,41.9,22
cleveland, oh,41.5,25.7
denver, co,39.7,29.2
honolulu, hi,21.3,73
jackson, ms,32.3,45
knoxville, tn,36,37.6
las vegas, nv,36.2,47

Explanation:

Step1: Predict fiber content for 3 - gram sugar cereal

Substitute $x = 3$ into $\hat{y}=-0.418x + 7.93$.
$\hat{y}=-0.418\times3 + 7.93$
$=-1.254+7.93$
$=6.676$

Step2: Calculate residual for 9 - gram sugar and 6 - gram fiber cereal

First, find the predicted value when $x = 9$:
$\hat{y}=-0.418\times9 + 7.93$
$=-3.762+7.93$
$=4.168$
The residual is the actual value minus the predicted value. So, residual $=6 - 4.168 = 1.832$

Step3: General note on $r^{2}$ calculation

To calculate $r^{2}$, we would typically use statistical software or formulas based on sums - of - squares (total sum of squares $SST$, regression sum of squares $SSR$, and residual sum of squares $SSE$). $r^{2}=\frac{SSR}{SST}=1-\frac{SSE}{SST}$. In the context of the sugar - fiber relationship, $r^{2}$ represents the proportion of the variability in the fiber content that can be explained by the linear relationship with the sugar content. Without the full data set for sums - of - squares calculations, we can't give a numerical value here.

Answer:

1. Predicted fiber content for 3 - gram sugar cereal: $6.676$ grams
2. Residual for 9 - gram sugar and 6 - gram fiber cereal: $1.832$ grams
3. For $r^{2}$: Numerical value can't be calculated without full data for sums - of - squares, but it represents the proportion of variability in fiber content explained by the linear relationship with sugar content.