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A diabetic is interested in determining how the amount of aerobic exercise impacts his blood sugar. When his blood sugar reaches \(170 \mathrm{mg} / \mathrm{dL}\), he goes out for a run at a pace of 10 minutes per mile. On different days, he runs different distances and measures his blood sugar after completing his run. Note: The preferred blood sugar level is in the range of 80 to \(120 \mathrm{mg} / \mathrm{dL}\). Levels that are too low or too high are extremely dangerous. The data generated are given in the following table. $$ \begin{array}{l|rrrrrrrrrrrr} \hline \text { Distance (miles) } & 2 & 2 & 2.5 & 2.5 & 3 & 3 & 3.5 & 3.5 & 4 & 4 & 4.5 & 4.5 \\ \hline \text { Blood sugar (mg/dL) } & 136 & 146 & 131 & 125 & 120 & 116 & 104 & 95 & 85 & 94 & 83 & 75 \\ \hline \end{array} $$ a. Construct a scatter diagram for these data. Does the scatter diagram exhibit a linear relationship between distance run and blood sugar level? b. Find the predictive regression equation of blood sugar level on the distance run. c. Give a brief interpretation of the values of \(a\) and \(b\) calculated in part \(\underline{b}\). d. Plot the predictive regression line on the scatter diagram of part a and show the errors by drawing vertical lines between scatter points and the predictive regression line e. Calculate the predicted blood sugar level count after a run of \(3.1\) miles \((5\) kilometers) f. Estimate the blood sugar level after a 10 -mile run. Comment on this finding.

Step by step solution

01

Construct a Scatter Diagram

Firstly, a scatter diagram needs to be constructed to visualize the data. Plot the distance run (in miles) on the x-axis and blood glucose level (mg/dL) on the y-axis. Mark each pair of (Distance, Blood Sugar) as a point on the plot.

02

Observe Linear Relationship

Observe if the points in the scatter diagram appear to fall along a straight line. If they do, that would indicate a linear relationship between the distance run and the blood sugar level.

03

Calculating Predictive Regression Equation

Use the model \(y=ax+b\) to calculate the coefficients a (slope) and b (y-intercept), where y represents the blood sugar level and x represents the distance run. The slope represents the rate of change in blood sugar levels for every mile run, and the y-intercept is the predicted blood sugar level at 0 miles.

04

Interpretation of Values of a and b

The value of 'a' shows the rate of change in blood sugar levels per mile run - it's the slope of the line. 'b' indicates the predicted blood sugar level at the start of the run - it's the intercept of the line.

05

Plotting the Predictive Regression Line

Plot the predictive regression line using the equation from Step 3 on the same scatter diagram. Also, show the errors by drawing vertical lines from each data point to the regression line.

06

Predict Blood Sugar Level After a Run of 3.1 miles

Use the prediction equation from Step 3 and substitute x=3.1 to predict the blood sugar level after a run of 3.1 miles.

07

Estimate Blood Sugar Levels after a 10-mile Run

Again, use the prediction equation and substitute x=10 to estimate the blood sugar level after a 10-mile run. Discuss the implications of this finding.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Scatter Plot

A scatter plot is a type of graph used to visualize the relationship between two numerical variables. In this exercise, we are looking at how the distance run (in miles) affects blood sugar levels (measured in mg/dL). Plotting data points on the graph helps you see trends or patterns in your dataset.

For example, by placing the distance run on the x-axis and the corresponding blood sugar level on the y-axis, you can identify whether there seems to be a consistent pattern or trend.

• If the data points are all over the place with no discernable pattern, then there is likely no relationship.
• If they form a visible trend, like increasing or decreasing, that indicates a correlation.

In this context, the diabetic individual's data points can be visualized to produce an initial understanding of how their exercise impacts blood sugar.

Linear Relationship

In the context of a scatter plot, a linear relationship appears when data points approximate a straight line. This implies that the two variables change at a constant rate relative to each other. In our example, we're examining if the increase in distance run results in a consistent decrease in blood sugar.

A linear relationship can be spotted if the data points align closely around an imaginary straight line through them. This would mean that as the distance increases, the blood sugar likely falls, creating a negative slope. If the points are linear, it's easier to create a predictive model, as we'll discuss shortly.

Detecting a linear relationship is crucial because it justifies using linear regression to develop a model that accurately reflects real-world phenomena.

Predictive Model

A predictive model leverages mathematical equations to forecast future values based on historical data. Here, we're focusing on predicting blood sugar levels based on the distance run. This involves creating a regression equation in the form of \( y = ax + b \), where \( y \) is the blood sugar level and \( x \) is the distance run.

In this equation:

• \( a \) is the slope, indicating how much blood sugar levels change per mile.
• \( b \) is the y-intercept, showing the expected blood sugar level without any running.

This model is very useful because if a linear relationship exists, one can predict blood sugar for distances not explicitly measured, simply by plugging the distance into this equation.

It's essential to remember that while predictive models can provide relatively accurate estimates, they are based on assumptions that might not hold for every scenario.

Blood Sugar Level Prediction

Blood sugar level prediction allows individuals, especially those with diabetes, to manage their health more effectively. Using the predictive model we've discussed, one can calculate expected blood sugar levels after running different distances.

For instance, you can insert 3.1 miles into the regression equation to predict the blood sugar level for a person running that distance. Similarly, estimating the blood sugar level after a 10-mile run can guide further health and exercise decisions.

• This prediction helps in planning the intensity and distance of exercises to maintain optimal blood sugar levels.
• It helps prevent dangerous hypo- or hyperglycemic episodes by foreseeing how exercise influences blood sugar.

Overall, accurately predicting blood sugar based on running distances can greatly aid in maintaining safe and healthy levels, thus supporting better diabetes management.