Problem 72 - A Graphical Approach to College Algebra

Question

The table lists the worldwide average household spending (in dollars) on Apple products for selected years. $$\begin{array}{|l|c|c|c|c|} \hline \text { Year } & 2009 & 2011 & 2013 & 2015 \\\ \hline \begin{array}{c} \text { Spending } \\ \text { (\$ dollars) } \end{array} & 62 & 158 & 265 & 444 \end{array}$$ (a) Use regression to find a formula \(f(x)=a x+b\) so that \(f\) models the data. (b) Interpret the slope of the graph of \(y=f(x)\) (c) Estimate the average household spending on Apple products in 2014 and compare it with the actual value of \(\$ 343\)

Step-by-Step Solution

Answer

The regression equation is \( f(x) = 33.975x + 130.325 \); spending increases by \$34 yearly. Estimated 2014 spending is \$299.2.

Step 1: Identify Variables

To solve for the regression line, identify the independent variable as the year (denoted as \( x \)) and the dependent variable as the spending (denoted as \( y \)). To simplify, let \( x = 0 \) correspond to the year 2009, so \( x \) will be the number of years since 2009.

Step 2: Calculate Regression Coefficients

Use the least squares method to calculate the regression coefficients \( a \) (slope) and \( b \) (intercept). We need values \( \bar{x} \) and \( \bar{y} \):1. \( \bar{x} = \frac{0 + 2 + 4 + 6}{4} = 3 \)2. \( \bar{y} = \frac{62 + 158 + 265 + 444}{4} = 232.25 \)Using formulas for slope \( a = \frac{\sum{(x_i - \bar{x})(y_i - \bar{y})}}{\sum{(x_i - \bar{x})^2}} \) and intercept \( b = \bar{y} - a \bar{x} \), we calculate these values.

Step 3: Calculate Slope (a)

Compute the slope \( a \):\[ a = \frac{(0-3)(62-232.25) + (2-3)(158-232.25) + (4-3)(265-232.25) + (6-3)(444-232.25)}{(0-3)^2 + (2-3)^2 + (4-3)^2 + (6-3)^2} \]This simplifies to:\[ a = \frac{(0)(-170.25) + (-2)(-74.25) + (1)(32.75) + (3)(211.75)}{9 + 1 + 1 + 9} = \frac{679.5}{20} \approx 33.975 \]

Step 4: Calculate Intercept (b)

Using the values obtained, calculate the intercept \( b \):\[ b = 232.25 - 33.975 \times 3 \approx 130.325 \]

Step 5: Formulate Regression Equation

With \( a \approx 33.975 \) and \( b \approx 130.325 \), the regression function is:\[ f(x) = 33.975x + 130.325 \]

Step 6: Interpret the Slope

The slope \( 33.975 \) indicates that the average household spending on Apple products increases by approximately \$34 per year.

Step 7: Estimate Spending for 2014

Substitute \( x = 5 \) (since 2014 is 5 years after 2009) into the formula:\[ f(5) = 33.975 \times 5 + 130.325 = 299.2 \] dollars. Compare this estimate with the actual spending value of \$343.

Key Concepts

Slope Interpretation

When discussing linear regression, the slope is a crucial component. It helps us understand the relationship between the two variables in question.
In our exercise, the slope, with a calculated value of approximately 33.975, tells us about the rate of change. Here, it means that for every year after 2009, the average household spending on Apple products increased by about $34.
This is a direct interpretation of how the slope in a linear equation like the one we have, \( f(x) = ax + b \), can describe real-world changes.

• The slope is positive, indicating an increase.
• The numerical value represents the average annual increment in spending.

Linear regressions help us gauge and assume patterns in data over a continuous interval.

Regression Equation

The regression equation is a representation that helps model the data in linear regression. In our exercise, this equation gradually evolved from the data to become \( f(x) = 33.975x + 130.325 \).
This equation delineates a line best fitting the given data points, which helps predict and analyze trends. Each part of the equation has a distinct role:

• \( a \) or the slope, describes how steep the line is and indicates the rate of change of spending over years.
• \( b \) or the y-intercept, in our calculation approximately 130.325, is the point where the line crosses the y-axis. It's the predicted spending in the initial year (when \( x = 0 \), or 2009).

By using this equation, we can estimate thousands of other data points in between the years provided, and even beyond, making it an extremely valuable analytical tool.

Household Spending Analysis

Analyzing household spending information with regression analyses offers crucial insights into consumer behavior.
In the problem at hand, the regression model tells us how the spending habit of households on Apple products has been steadily increasing over the years. When we analyze spending for any intervening years, like 2014, we use \( f(x) \) to predict:
Substituting 2014 into our equation, \( f(x) = 33.975 \times x + 130.325 \), where \( x = 5 \), we calculate the expected spending.

• This calculated estimate is 299.2 dollars.
• The actual spending, according to the exercise, was 343 dollars.

This discrepancy indicates the limits of a linear model; it may not account for all variables or deviations like increased Apple product releases or consumer trends in 2014. Nonetheless, it serves as a rough estimate for understanding overall trends across a time continuum.