Problem 72 - Applied Calculus

Question

Several surveys in the United States and Europe have asked people to rate their happiness on a scale of \(3=\) "very happy," \(2=\) "fairly happy," and \(1={ }^{\prime \prime}\) not too happy," and then tried to correlate the answer with the person's income. For those in one income group (making \(\$ 25,000\) to \(\$ 55,000\) ) it was found that their "happiness" was approximately given by \(y=0.065 x\) \(-0.613 .\) Find the reported "happiness" of a person with the following incomes (rounding your answers to one decimal place). a. \(\$ 25,000\) b. \(\$ 35,000\) c. \(\$ 45,000\)

Step-by-Step Solution

Answer

a. 1624.4, b. 2274.4, c. 2924.4

Step 1: Understanding the Equation

The given equation is the linear relation between happiness, denoted by \( y \), and income, denoted by \( x \). The equation provided is \( y = 0.065x - 0.613 \). Our task is to find the happiness rating for different income levels by substituting these income values into the equation for \( x \).

Step 2: Substitute and Calculate for $25,000

Substitute \( x = 25,000 \) into the equation. The equation becomes:\[y = 0.065(25000) - 0.613\]Calculate:\[y = 1625 - 0.613 = 1624.387\]Round to one decimal place: \( y = 1624.4 \).

Step 3: Substitute and Calculate for $35,000

Substitute \( x = 35,000 \) into the equation. The equation becomes:\[y = 0.065(35000) - 0.613\]Calculate:\[y = 2275 - 0.613 = 2274.387\]Round to one decimal place: \( y = 2274.4 \).

Step 4: Substitute and Calculate for $45,000

Substitute \( x = 45,000 \) into the equation. The equation becomes:\[y = 0.065(45000) - 0.613\]Calculate:\[y = 2925 - 0.613 = 2924.387\]Round to one decimal place: \( y = 2924.4 \).

Key Concepts

Happiness Index

The Happiness Index often seeks to measure how satisfied or happy individuals feel about their lives. These surveys usually ask participants to rate their happiness on a scale. In this study, participants have been asked to rate their happiness as "very happy," "fairly happy," or "not too happy," corresponding to values of 3, 2, and 1 respectively.
Understanding the Happiness Index is essential because it sheds light on the well-being of a population.
By correlating these happiness ratings with various factors, researchers aim to gain insights into what influences happiness.

• For example, factors such as income levels, employment status, and personal relationships are often analyzed.
• In this case, we're looking at the relationship between income levels and happiness.

This index helps governments, organizations, and societies understand what might improve lives comprehensively, beyond just financial measures.

Income Correlation

Income correlation refers to the degree to which income and another variable, like happiness, are related. A positive correlation indicates that as income increases, happiness increases, whereas a negative correlation would imply the opposite.
For our specific this exercise, the correlation between happiness and income is expressed through a linear equation:

• The equation given is: \( y = 0.065x - 0.613 \).
• Here, \( y \) represents the happiness score, and \( x \) is the income level.

This specific linear model suggests that there is a positive correlation between income and happiness, although the influence is relatively small, as indicated by the coefficient 0.065. The value -0.613 represents the hypothetical happiness score when income would be zero, often referred to as the y-intercept.

Linear Models in Statistics

Linear models are powerful tools in statistics used to describe the relationship between two variables. These equations predict outcomes based on a linear trend observed over a data set.
In our example, the linear model \( y = 0.065x - 0.613 \) describes how happiness (\( y \)) is expected to change with income (\( x \)).
Key characteristics of linear models include:

• The slope of the line: Here, 0.065, which signifies the rate of change in happiness per dollar increase in income.
• The intercept: -0.613 demonstrates the expected happiness level at an income of zero.

This simplistic model captures the overall trend between happiness and income in this defined income group. However, remember that other variables and non-linear factors can also influence happiness, which aren't captured in this straightforward model.

Applied Calculus

Applied calculus involves using derivatives and integrals to solve real-world problems, like optimizing economic outcomes or predicting trends. While this exercise primarily uses algebra, the principles of calculus subtly underpin the reasoning.

• For instance, the rate of change, captured by the slope 0.065 in our equation, is akin to a derivative, representing how one variable responds to changes in another.
• In extensive studies, calculus helps to understand the broad, ongoing changes rather than the specific, one-time calculations.

While you may not see the direct use of calculus in the initial steps, it forms the backbone of economic and statistical analyses concerning trends and rates of change, enhancing the computational models used to predict and understand complex systems.