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Chapter 3: Problem 30

Government debt and population \(\quad\) Data used in this exercise was published by www.bloomberg.com for the most government debt per person for 58 countries and their respective population sizes in 2014 . When using population size (in millions) as the explanatory variable \(x,\) and government debt per person (in dollars) as the response variable \(y,\) the regression equation is predicted as government debt per person \(=19560.405-13.495\) population. a. Interpret the slope of the regression equation. Is the association positive or negative? Explain what this means. b. Predict government debt per person at the (i) minimum population size \(x\) value of 4 million, (ii) at the maximum population size \(x\) value of 1367.5 million. c. For India, government debt per person \(=\$ 946,\) and population \(=1259.7\) million. Find the predicted gov. ernment debt per person and the residual for India. Interpret the value of this residual.

Short Answer

Expert verified
The slope is -13.495, indicating a negative relationship. Predicted government debts are $19,506.43 for 4 million and $1,119.29 for 1367.5 million populations. India's residual of -$1,624.95 suggests its lower actual debt per person.

Step by step solution

01

Interpreting the Slope

The slope of the regression equation is -13.495. This indicates that for every increase of 1 million in population size, the government debt per person decreases by approximately $13.495. Since the slope is negative, the association between population size and government debt per person is negative.
02

Predicting Government Debt for 4 Million Population

Using the regression equation, plug in 4 for the population size:\[y = 19560.405 - 13.495 imes 4\]Calculate \(y\):\[y = 19560.405 - 53.98 = 19506.425\]Thus, the predicted government debt per person for a population of 4 million is $19,506.43.
03

Predicting Government Debt for 1367.5 Million Population

Using the regression equation, plug in 1367.5 for the population size:\[y = 19560.405 - 13.495 imes 1367.5\]Calculate \(y\):\[y = 19560.405 - 18441.1125 = 1119.2925\]Thus, the predicted government debt per person for a population of 1367.5 million is $1,119.29.
04

Finding Predicted Debt for India

Using the regression equation, plug in 1259.7 for the population size:\[y = 19560.405 - 13.495 imes 1259.7\]Calculate \(y\):\[y = 19560.405 - 16989.4515 = 2570.9535\]Thus, the predicted government debt per person for India's population is approximately $2,570.95.
05

Calculating the Residual for India

The residual is the difference between the observed and predicted government debt per person. For India, the observed debt is \(946.\[\text{Residual} = 946 - 2570.9535 = -1624.9535\]The negative residual means India's actual government debt per person is \)1,624.95 less than predicted.
06

Interpreting the Residual

The negative residual indicates that India's government debt per person is significantly lower than what the regression model predicts. This could suggest that, relative to its population, India has managed its debt per person better than other countries in the analyzed data set.

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Key Concepts

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

Slope Interpretation
In regression analysis, the slope of the equation reveals the relationship between the explanatory and response variables. Let's break it down using the regression equation provided: - The slope in the equation, -13.495, tells us how much the government debt per person changes when the population size increases by one unit (or in this case, by 1 million people). - With a negative slope, as we see here, there's an inverse relationship. It means that for every increase of 1 million people in population size, the government debt per person decreases by approximately $13.495. This negative association conveys an essential insight that larger populations tend to have lesser government debt per person, according to this model. Understanding the slope helps us predict how changes in population size might inversely affect government debt per person in different scenarios.
Predictive Modeling
Predictive modeling, in this context, refers to using the regression equation to make predictions about government debt per person for different population sizes. Here's how it operates:- By substituting a specific population value (\(x\)) into the equation, we can compute an expected value for government debt per person (\(y\)).- For example, with a population size of 4 million, we plug the number into the equation: \[y = 19560.405 - 13.495 \times 4 \] which results in a predicted debt of \(19,506.43 per person.- Similarly, for a massive population size like 1367.5 million, the equation \[y = 19560.405 - 13.495 \times 1367.5 \] gives us a prediction of \)1,119.29 per person.Predictive modeling is powerful as it allows us to foresee potential outcomes and make informed decisions based on predicted data patterns.
Residual Calculation
Residual calculation is crucial for understanding the accuracy of predictions made by a regression model. Here's its role in this analysis:- The residual is the difference between the observed value and the predicted value of government debt per person. - For India, where the actual government debt per person is \(946, the predicted debt from the model is approximately \)2,570.95.- Calculating the residual, \[ ext{Residual} = 946 - 2570.95 = -1624.95 \]reveals that the model's prediction was higher by $1,624.95 than the actual amount.A negative residual, as seen here, suggests that the actual debt is less than the predicted debt, indicating a better economic situation than expected by the model's trends. Residuals help identify discrepancies and improve model reliability where needed.

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Most popular questions from this chapter

Predicting cost of meal from rating Refer to the previous exercise. The correlation with the cost of a dinner is 0.68 for food quality rating, 0.69 for service rating, and 0.56 for d茅cor rating. According to the definition of \(r^{2}\) as a measure for the reduction in the prediction error, which of these three ratings can be used to make the most accurate predictions for the cost of a dinner: quality of food, service, or d茅cor? Why?

For the following pairs of variables, which more naturally is the response variable and which is the explanatory variable? a. Carat ( \(=\) weight ) and price of a diamond b. Dosage (low/medium/high) and severity of adverse event (mild/moderate/strong/serious) of a drug c. Top speed and construction type (wood or steel) of a roller coaster d. Type of college (private/public) and graduation rate

Explain what's wrong with the way regression is used in each of the following examples: a. Winning times in the Boston marathon (at www. bostonmarathon.org) have followed a straight-line decreasing trend from 160 minutes in 1927 (when the race was first run at the Olympic distance of about 26 miles) to 128 minutes in 2014 . After fitting a regression line to the winning times, you use the equation to predict that the winning time in the year 2300 will be about 13 minutes. b. Using data for several cities on \(x=\%\) of residents with a college education and \(y=\) median price of home, you get a strong positive correlation. You conclude that having a college education causes you to be more likely to buy an expensive house. c. A regression between \(x=\) number of years of education and \(y=\) annual income for 100 people shows a modest positive trend, except for one person who dropped out after 10 th grade but is now a multimillionaire. It's wrong to ignore any of the data, so we should report all results including this point. For this data, the correlation \(r=-0.28\).

A study conducted among sophomores at Fairfield University showed a correlation of -0.69 between the number of hours spent watching TV and GPA. It was found that for each additional hour a week you spent watching \(\mathrm{TV}\), you could expect on average to see a drop of \(.0452,\) on a 4.0 scale, in your GPA. a. Is there a causal relationship, whereby watching more TV decreases your GPA? b. Explain how a student's intelligence, measured by her/ his IQ score, could be a lurking variable that might be responsible for this association, having a common correlation with both GPA and TV-watching hours.

According to data obtained from the General Social Survey (GSS) in 2014,1644 out of 2532 respondents were female and interviewed in person, 551 were male and interviewed in person, 320 were female and interviewed over the phone and 17 were male and interviewed over the phone. a. Explain how we could regard either variable (gender of respondent, interview type) as a response variable. b. Display the data as a contingency table, labeling the variables and the categories. c. Find the conditional proportions that treat interview type as the response variable and gender as the explanatory variable. Interpret. d. Find the conditional proportions that treat gender as the response variable and interview type as the explanatory variable. Interpret. e. Find the marginal proportion of respondents who (i) are female, (ii) were interviewed in person.
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