Problem 117 - Precalculus Essentials

Question

In Exercises \(87-88,\) we used the data in a bar graph to develop linear functions that modeled the percentage of never-married American females and males, ages \(25-29\). For this group exercise, you might find it helpful to pattern your work after Exercises 87 and \(88 .\) Group members should begin by consulting an almanac, newspaper, magazine, or the Internet to find data that appear to lie approximately on or near a line. Working by hand or using a graphing utility, group members should construct scatter plots for the data that were assembled. If working by hand, draw a line that approximately fits the data in each scatter plot and then write its equation as a function in slope-intercept form. If using a graphing utility, obtain the equation of each regression line. Then use each linear function's equation to make predictions about what might occur in the future. Are there circumstances that might affect the accuracy of the prediction? List some of these circumstances.

Step-by-Step Solution

Answer

The exercise doesn't have fixed data, hence a specific short answer can't be provided. However, the final result should consist of a linear function's equation used to make future predictions along with a list of potential factors that could affect the accuracy of these predictions.

Step 1: Data Identification and Collection

Start by finding relevant data which seem to follow a linear trend from sources like books, newspapers, the internet or magazines. Keep track of the source, as well as the variable in question.

Step 2: Creation of Scatter Plot

After gathering the data, construct a scatter plot where each point represents an observation from the collected data. The variable you chose should be represented on the x-axis and the corresponding data on the y-axis.

Step 3: Approximate Straight Line

When working manually, draw a straight line that approximately fits the plotted data points. If using a graphing utility, apply the linear regression function to get the line that best fits the data.

Step 4: Formulate Linear Function's Equation

Write the equation of the line in y=ax+b form, where 'a' represents the slope of the line and 'b' is the y-intercept. If using a graphing utility, it will provide the equation.

Step 5: Make Future Predictions

With the equation of the line, predict future outcomes. Substitute the value of the future variable into the x-value of the equation to obtain the predicted y-value.

Step 6: Acknowledge Limitations

Reflection on potential factors that may affect the accuracy of the model is necessary. List all the circumstances that could affect your predictions, such as outliers in your data set or drastic potential changes in the variables being tracked. Remember, predictions based on linear functions won't always be accurate for all situations, especially in the long-term.