Problem 72 - A Graphical Approach to Precalculus with Limits

Question

The table lists the worldwide average household spending (in dollars) on Apple products for selected years. $$\begin{array}{|l|c|c|c|c|} \hline \text { Year } & 2009 & 2011 & 2013 & 2015 \\ \hline \begin{array}{l} \text { Spending } \\ \text { (in dollars) } \end{array} & 62 & 158 & 265 & 444 \\ \hline \end{array}$$ (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

(a) Linear equation: \(f(x) = 63.67x + 62\). (b) Slope is increase by \$63.67 every two years. (c) Estimate for 2014: \$380.35, actual: \$343.

Step 1: Define Variables

Let's define the variables for the linear regression. Let \(x\) be the year and \(y\) be the spending in dollars. To simplify calculations, use \(x = 0, 2, 4, 6\) for years 2009, 2011, 2013, and 2015 respectively.

Step 2: Set Up the Linear Regression Formula

We use one point as \((x_1, y_1) = (0, 62)\) and another as \((x_2, y_2) = (6, 444)\) to calculate the slope \(a\) of the line: \(a = \frac{y_2 - y_1}{x_2 - x_1}\). Once we have \(a\), we calculate \(b\) using \(b = y_1 - a \cdot x_1\).

Step 3: Calculate the Slope \(a\)

The slope is \(a = \frac{444 - 62}{6 - 0} = \frac{382}{6} \approx 63.67\).

Step 4: Calculate the Intercept \(b\)

Using \(b = y_1 - a \cdot x_1\), substitute \(y_1 = 62\) and \(x_1 = 0\) to get \(b = 62 - 63.67 \times 0 = 62\). Thus, the equation becomes \(f(x) = 63.67x + 62\).

Step 5: Interpret the Slope

The slope \(a = 63.67\) represents that, on average, household spending on Apple products increases by approximately \$63.67 every two years.

Step 6: Estimate Spending in 2014

Using the year 2014, which corresponds to \(x = 5\), substitute into the regression function \(f(x) = 63.67x + 62\): \[f(5) = 63.67 \times 5 + 62 \approx 380.35\].

Step 7: Compare Estimated and Actual Value for 2014

The estimated spending for 2014 is \\(380.35, which is higher than the actual spending of \\)343.

Key Concepts

Slope Interpretation

When we talk about the slope in linear regression, we're really discussing how much one variable changes in relation to another. In our exercise, the slope tells us how household spending on Apple products changes as time goes on.

Here's the key point: the slope we calculated is approximately 63.67. What does this mean?

• It indicates that, on average, spending increases by $63.67 for every 2-year interval. So, if you imagine a line graph representing the data, the slope describes how steeply that line rises as years go by.
• A positive slope shows an upward trend, which in this context suggests that people tend to spend more on Apple products over time.
• If the slope were negative, it would indicate a decrease in spending as years progress, but that's not the case here.

Understanding slope is crucial for making predictions about future spending patterns, as it gives us insight into the rate of change over time.

It’s like reading between the lines, literally! This knowledge helps in forecasting budgets and planning future expenses.

Data Modeling

In data modeling, you're essentially creating a math-based representation of the real-world scenario. This involves using data points to find the best fit equation which helps in making predictions.

For the exercise above, we built a linear model using the equation \(f(x) = 63.67x + 62\). Here's how this works:

• The variable \(x\) represents the time measured in years. In this model, we simplified our calculations by using integers for years (2009 becomes 0, 2011 becomes 2, etc.).
• Our model, \(f(x)\), is a straight line that estimates the household spending on Apple products based on these years. "Fitting" means finding the best line that represents the data we have.

The process includes calculating the slope and intercept to define the line.

This model assumes that the relationship between time and spending is linear, meaning spending will continually increase at a constant rate. But remember, real-world data can show some variabilities.

Thus, while data modeling is immensely helpful in predictions and analysis, it's also important to be aware of its limitations due to assumptions and the nature of the data.

Yearly Spending Analysis

Yearly spending analysis helps us understand how spending or resource allocation changes over time. With the data and the linear model at hand, we predicted the average spending in 2014.

Here's what we did: we plugged the year 2014 into the function \(f(x) = 63.67x + 62\), with \(x\) reprising as 5 (since 2014 is five years past 2009):

• Calculation: \(f(5) = 63.67 \times 5 + 62 = 380.35\).
• This suggests an expected spending of \(380.35 in 2014.

Comparing this to the actual spending of \)343, it’s clear our model slightly overestimated the spending. What does this reveal?

• It highlights the importance of understanding models are simplified versions of reality. They are not always perfectly accurate.
• Analyzing these disparities can offer insights into unexpected factors affecting spending in those years.

Conducting a yearly spending analysis is, thus, a crucial aspect of financial planning and economic studies as it helps uncover trends and refine current models.