Question

unit 2 - exploring two variable data hw 3 - linear regression name: lucy lu 1) fiber helps regulate the body’s use of sugars, helping to keep hunger and blood sugar in check. in children’s cereal, which is usually loaded with large amounts of sugar, scientists wanted to investigate if a larger sugar content led to less nutritional value overall, such as lower fiber. a scatter plot of 15 randomly selected children’s cereals, along with some selected summary statistics, are given below. both sugar and fiber are measured in grams per serving. a) does the scatterplot indicate that it is okay to create a linear model? explain. yes. it b) find the slope of the least - squares regression line. interpret this value in context. c) what point must be on the least - squares regression line? d) find the intercept of the least - squares regression line. interpret this value in context.

Explanation:

Step1: Check linear - model suitability

The points in the scatter - plot show a somewhat linear trend. Although not perfectly linear, there is no strong curvature, so it is reasonable to create a linear model.

Step2: Calculate the slope formula

The formula for the slope $b_1$ of the least - squares regression line is $b_1=r\frac{s_y}{s_x}$, where $r$ is the correlation coefficient, $s_y$ is the standard deviation of the response variable, and $s_x$ is the standard deviation of the explanatory variable.
Given $r = - 0.654$, $s_x=5.18$, and $s_y = 3.31$.
$b_1=-0.654\times\frac{3.31}{5.18}$
$b_1=-0.654\times0.639$
$b_1\approx - 0.418$
Interpretation: For every one - gram increase in sugar content, the fiber content is expected to decrease by approximately $0.418$ grams.

Step3: Identify the point on the regression line

The point $(\bar{x},\bar{y})$ must be on the least - squares regression line. Here, $\bar{x}=8.6$ and $\bar{y}=4.33$, so the point is $(8.6,4.33)$.

Step4: Calculate the intercept formula

The formula for the intercept $b_0$ of the least - squares regression line is $b_0=\bar{y}-b_1\bar{x}$.
We know $\bar{x}=8.6$, $\bar{y}=4.33$, and $b_1\approx - 0.418$.
$b_0=4.33-(-0.418)\times8.6$
$b_0=4.33 + 3.605$
$b_0\approx7.935$

Answer:

a) Yes. The points show a somewhat linear trend with no strong curvature.
b) Slope: $b_1\approx - 0.418$. For every one - gram increase in sugar content, the fiber content is expected to decrease by approximately $0.418$ grams.
c) $(8.6,4.33)$
d) Intercept: $b_0\approx7.935$