91Ó°ÊÓ

The previous problem discusses GDP, which is a commonly used measure of the overall economic activity of a nation. For this group of nations, the GDP data have a mean of 1909 and a standard deviation of 3136 (in billions of U.S. dollars). a. The five-number summary of GDP is minimum \(=204\), \(\mathrm{Q} 1=378,\) median \(=780, \mathrm{Q} 3=2015,\) and maximum \(=16,245 .\) Sketch a box plot. b. Based on these statistics and the graph in part a, describe the shape of the distribution of GDP values. c. The data set also contains per capita GDP, or the overall GDP divided by the nation's population size. Construct a scatterplot of per capita GDP and GDP and explain why no clear trend emerges. d. Your friend, Joe, argues that the correlation between the two variables must be 1 since they are both measuring the same thing. In reality, the actual correlation between per capita GDP and GDP is only \(0.32 .\) Identify the flaw in Joe's reasoning.

Step by step solution

01

Identify the Elements for the Box Plot

To sketch a box plot, we need the five-number summary, which includes: - Minimum = 204 - First Quartile (Q1) = 378 - Median = 780 - Third Quartile (Q3) = 2015 - Maximum = 16,245

02

Sketch the Box Plot

Draw a number line that includes the range of your data (from 0 to around 17,000). Next, plot the five points of the summary on this line and draw a box from Q1 (378) to Q3 (2015) with a line at the median (780). Then, extend 'whiskers' from the box to the minimum (204) and maximum (16,245) values. Note that the length of the whiskers and the box suggest the distribution's skew.

03

Describe the Shape of the Distribution

In observing the box plot, the lengthy right whisker and large maximum point compared to other values indicate that the GDP distribution is right-skewed. This means there are some nations with extremely high GDP that shift the distribution to the right.

04

Explain the Absence of a Trend in Scatterplot

In a scatterplot of per capita GDP versus total GDP, each point represents the GDP of a nation relative to its population size. However, because GDP can be high due to either a large population or high economic output per person, no straightforward trend will emerge. Nations with similar GDP can have different population sizes and economic productivity rates, scattering points widely on the plot.

05

Detail the Flaw in Joe's Reasoning

Joe's mistake stems from misunderstanding correlation. While GDP and per capita GDP relate, they measure different aspects — overall economic output vs. economic output per person. Per capita figures vary widely based on population sizes and economic structures, hence they correlate less strongly than Joe expects, leading to a correlation of only 0.32 rather than 1.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Box Plot

A box plot is a clever graphical tool that helps you visualize the distribution of a data set. It's like a quick snapshot!
To create a box plot, you start with the five-number summary, which includes the minimum, first quartile (Q1), median, third quartile (Q3), and maximum values.

• Minimum: The smallest value— in our case, it's 204.
• Q1: This is the median of the lower half of the data, at 378.
• Median: The middle value, which here is 780.
• Q3: The median of the upper half, which is 2015.
• Maximum: The largest value, which is 16,245 in this example.

Draw a number line that covers the spread of your data, from 0 to about 17,000 for this GDP data. Plot these five points on the line and create a box from Q1 to Q3 with a line at the median. This box shows where the bulk of your data lies.
Extend whiskers from the box to the minimum and maximum values. The length of the whiskers and the box can also suggest if the distribution is skewed.

Scatterplot

A scatterplot is a great way to visualize the relationship between two variables. In this case, we're examining GDP and per capita GDP. Each point on the plot represents a country, with its GDP on one axis and its per capita GDP on the other. This helps us see if there is a trend or pattern. Scatterplots can sometimes show a clear trend where, as one variable increases, so does the other. But in our case, no obvious pattern pops out.
Why is that?
The scatterplot here doesn't show a distinct trend because a country's GDP can be influenced by several factors, like the size of its population and the economic output per person. What this means, is that two countries could have similar GDPs but vastly different population sizes or economic productivity levels. This can lead to points being scattered across the plot without forming a clear line or curve.

Correlation

Correlation helps us understand the strength and direction of a relationship between two variables. Think of it as our measure of the dance between them! A perfect correlation of 1 means they move in perfect sync, while a correlation of 0 means there's no relationship at all. A correlation of -1 indicates a perfect inverse relationship—when one goes up, the other goes down.
In our case, the correlation between GDP and per capita GDP is 0.32. This positive but low correlation suggests a tiny relationship. Essentially, this means that changes in GDP do not strongly predict changes in per capita GDP, or vice versa. Joe's assumption that the correlation should be 1 is flawed. Although GDP and per capita GDP are related, they measure different things. GDP measures total economic output, while per capita GDP reflects economic output per individual. People often confuse this, leading to exaggerated expectations of correlation.

Right-Skewed Distribution

A right-skewed distribution, sometimes called positively skewed, means that more of your data points fall on the left of the distribution, and there's a longer tail extending to the right. This can often indicate a few unusually high data points. In our GDP data, you see this clearly from the box plot. The long whisker on the right side, coupled with the large maximum point, signals a right-skew.
What causes the skewness in this distribution? A small number of nations have much higher GDPs than the rest, which shifts the entire distribution to the right. This kind of skewness is important to recognize as it can impact the way you analyze data. For instance, in this case, it shows us the presence of wealthy countries with GDPs that are outliers compared to most others. This is crucial when interpreting averages since a few high values can inflate the average significantly.