Question

In Exercise 2.187 on page 118 , we introduce the dataset HappyPlanetIndex, which includes information for 143 countries to produce a "happiness" rating as a score of the health and well-being of the country's citizens, as well as information on the ecological footprint of the country. One of the variables used to create the happiness rating is life expectancy in years. We explore here how well this variable, LifeExpectancy, predicts the happiness rating, Happiness. (a) Using technology and the data in HappyPlanetIndex, create a scatterplot to use LifeExpectancy to predict Happiness. Is there enough of a linear trend so that it is reasonable to construct a regression line? (b) Find a formula for the regression line and display the line on the scatterplot. (c) Interpret the slope of the regression line in context.

Short answer

The scatterplot and linear regression model are tools for understanding the relationship between life expectancy and happiness. Declaring that life expectancy can predict happiness depends primarily on the slope of the regression line, which indicates the strength and direction of the connection between the two variables. Its interpretation should contain a practical and contextual explanation.

Step-by-step solution

Create a Scatterplot

Using software like Excel or a statistical computing language like R or Python, input the dataset from HappyPlanetIndex. Once inputted, plot a scatterplot with LifeExpectancy as the independent variable (x-axis) and Happiness as the dependent variable (y-axis). Then study the scatterplot for any linear trends.

Construct a Regression Line

If there is a linear trend between the two variables (LifeExpectancy and Happiness), decompose the regression model. The regression line equation will be in the form \(Y = BX + A + \epsilon\), where Y is the predicted happiness rating, B is the slope of the regression line, X is the life expectancy, A is the y-intercept, and \(\epsilon\) is the error term. Use the technology to compute B and A.

Display the Regression Line

Once the regression line's formula is obtained, add the regression line to the existing scatterplot. The line represents the best estimate of happiness based on life expectancy.

Interpret the Slope

The slope of the regression line (B) represents the amount of change in the predicted happiness rating for each one-unit increase in life expectancy. In practical terms, if B is positive, it means that life expectancy and happiness have a direct relationship: as life expectancy increases, happiness also increases. If B is negative, it implies an inverse relationship: as life expectancy increases, happiness decreases.

Key concepts

Scatterplot

Imagine a graph where you want to check if there's a connection between how long people live in a country and how happy they feel. That's where a scatterplot comes in handy. It's a type of graph that shows points on a horizontal and a vertical line. Each point represents two things – like the number of years people expect to live, and how happy they are.

A scatterplot can show you at a quick glance if these two things might be related. For example, if you see most points going up as they move to the right, there might be a pattern where living longer equals being happier. But if the points are all over the place with no clear line, then maybe there's no strong link between the two.

Happiness Rating

Now, 'happiness rating' is a score that tries to measure how well people feel about their life in their country. It's like giving happiness a number, so we can compare different countries. It's tricky because happiness can be about so many things like health, money, freedom, or even how clean the air is! When we study happiness ratings, we're looking for what might make people feel better about their lives.

Life Expectancy

Life expectancy is the average number of years people are expected to live in a country. It's a big clue about the health of a place. More years usually means people have things like good food, healthcare, and a safe place to live. It's why many researchers believe that life expectancy and happiness might be connected—the longer you live, the better life might be, which could make you happier.

Regression Line

Think of a regression line like a magic wand that tries to make sense of all the scattered points. It's a straight line that best fits through the points on the scatterplot. This line tries to show us the relationship between life expectancy and happiness. It's not perfect, but it's the best straight-line guess for where the points tend to go. If life expectancy is on the x-axis and happiness on the y-axis, then this line tells us what happiness we might expect for a given number of living years. And the way we find this line is by using some math magic called 'regression analysis'.

Slope Interpretation

In the world of graphs, 'slope' is a term for how steep a line is. In our special graph, it tells us how much happiness might increase (or decrease) for each extra year of life expectancy. If the slope is going up, that means more years could equal more happy feelings. The steeper the slope, the stronger this effect. But if it's going down, it's saying the opposite, that more years aren't bringing more happiness. This slope number is super important because it gives us a hint—it's not just if, but how much life expectancy might affect happiness.