Question

e) write the equation of the linear model.
f) if you pick up a cereal and see that it has 3 grams of sugar, what is the predicted fiber content?
g) what is the residual for the cereal with 9 grams of sugar and 6 grams of fiber?
h) calculate the value of r² and interpret this value in context.

1. 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 scatterplot are given below.

city,latitude (°n),dec average temp (°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: Identify the problem type

This problem involves creating a linear - model, making predictions, calculating residuals, and interpreting the coefficient of determination ($r^{2}$) based on given data. It is a statistics - related problem.

Step2: Assume the linear model form

The general form of a simple linear regression model is $\hat{y}=b_{0}+b_{1}x$, where $\hat{y}$ is the predicted value, $b_{0}$ is the y - intercept, $b_{1}$ is the slope, and $x$ is the independent variable.

Step3: Calculate the slope ($b_{1}$) and y - intercept ($b_{0}$)

The formulas for $b_{1}$ and $b_{0}$ are:
$b_{1}=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}$ and $b_{0}=\bar{y}-b_{1}\bar{x}$, where $\bar{x}$ and $\bar{y}$ are the means of the independent and dependent variables respectively, and $n$ is the number of data points.

Step4: For part (f)

Once we have the linear model $\hat{y}=b_{0}+b_{1}x$, if $x = 3$ (grams of sugar), we substitute $x = 3$ into the model to get the predicted fiber content $\hat{y}$.

Step5: For part (g)

The residual $e$ is calculated as $e=y-\hat{y}$. Given $x = 9$ (grams of sugar) and $y = 6$ (grams of fiber), we first find $\hat{y}$ using the linear model with $x = 9$, then $e=6-\hat{y}$.

Step6: For part (h)

The coefficient of determination $r^{2}$ is calculated as $r^{2}=1-\frac{SSE}{SST}$, where $SSE=\sum_{i = 1}^{n}(y_{i}-\hat{y}_{i})^{2}$ (sum of squared errors) and $SST=\sum_{i = 1}^{n}(y_{i}-\bar{y})^{2}$ (total sum of squares). $r^{2}$ represents the proportion of the variance in the dependent variable that is predictable from the independent variable.

However, since no actual calculations are done here (as the data for the sugar - fiber relationship is not fully given in a calculable form), we will assume a general approach.

Answer:

For part (f): Substitute $x = 3$ into the linear model $\hat{y}=b_{0}+b_{1}x$ to get the predicted fiber content.
For part (g): First find $\hat{y}$ for $x = 9$ using the linear model, then calculate the residual $e = 6-\hat{y}$.
For part (h): Calculate $r^{2}=1-\frac{SSE}{SST}$, where $SSE=\sum_{i = 1}^{n}(y_{i}-\hat{y}_{i})^{2}$ and $SST=\sum_{i = 1}^{n}(y_{i}-\bar{y})^{2}$. $r^{2}$ represents the proportion of the variance in the dependent variable that can be explained by the independent variable.