/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 30 Data used in this exercise was p... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 30

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 government debt per person and the residual for India. Interpret the value of this residual.

Short Answer

Expert verified
a. Negative; it means debt per person decreases as population increases. b. (i) $19506.425, (ii) $1127.6925. c. Predicted: $2565.7035, Residual: $-1619.7035; actual is lower than predicted.

Step by step solution

01

Understand the Regression Equation

The regression equation given is: \[\text{government debt per person} = 19560.405 - 13.495 \times \text{population size (in millions)}\] This tells us how government debt per person changes as the population size changes.
02

Interpret the Slope

The slope of the equation is \(-13.495\). This means that for each increase of 1 million in population size, the government debt per person decreases by approximately \(13.495\) dollars. The association is negative because of the negative sign of the slope.
03

Predict for Minimum Population Size

Substitute \(x = 4\) into the regression equation:\[y = 19560.405 - 13.495 \times 4\]Calculate:\[y = 19560.405 - 53.98 = 19506.425\]Thus, the predicted government debt per person for a population of 4 million is approximately \(19506.425\) dollars.
04

Predict for Maximum Population Size

Substitute \(x = 1367.5\) into the regression equation:\[y = 19560.405 - 13.495 \times 1367.5\]Calculate:\[y = 19560.405 - 18432.7125 = 1127.6925\]Thus, the predicted government debt per person for a population of 1367.5 million is approximately \(1127.6925\) dollars.
05

Predicted Government Debt for India

Substitute \(x = 1259.7\) into the regression equation:\[y = 19560.405 - 13.495 \times 1259.7\]Calculate:\[y = 19560.405 - 16994.7015 = 2565.7035\]The predicted government debt per person for India is approximately \(2565.7035\) dollars.
06

Residual Calculation for India

The actual government debt per person for India is \(946\) dollars. The predicted value is \(2565.7035\) dollars. The residual is calculated as:\[\text{Residual} = \text{Actual} - \text{Predicted} = 946 - 2565.7035 = -1619.7035\]The residual of \(-1619.7035\) dollars means the actual government debt per person in India is \(1619.7035\) dollars less than what the model predicts.

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Key Concepts

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

Slope Interpretation
In the realm of linear regression, the slope of the regression line is crucial for understanding how one variable changes in relation to another. In this particular exercise, the regression equation is presented as follows:
\[\text{Government debt per person} = 19560.405 - 13.495 \times \text{Population size (in millions)}\]
The slope here is \(-13.495\), which signifies a negative association between the population size and the government debt per person.
  • A negative slope indicates that as the population size increases by 1 million, the government debt per person decreases by approximately 13.495 dollars.
  • This inverse relationship suggests that larger populations tend to result in lower government debt per capita, according to the model.
It's important to consider the context: while the model demonstrates a mathematical relationship, real-world factors might influence these numbers in ways not captured by the regression equation alone.
Residual Calculation
Residuals are a valuable component of regression analysis as they reveal how well the regression model predicts actual data points. Calculating residuals involves comparing the actual observed values to those predicted by the regression model.
In this exercise, for instance, the actual government debt per person for India is given as \(946\) dollars, while the regression model predicts \(2565.7035\) dollars.
  • The residual is calculated as:
    \[\text{Residual} = \text{Actual} - \text{Predicted} = 946 - 2565.7035 = -1619.7035\]
  • This negative residual value of \(-1619.7035\) dollars indicates that the observed debt is substantially less than what the model expected.
Residuals help identify how closely a model's predictions match actual outcomes, and significant residuals might suggest that other factors are affecting the data. A large residual can hint at limitations in the model.
Predictive Analysis
Predictive analysis in linear regression involves using the established regression line to predict future outcomes based on given data points. This exercise exemplifies how predictions are made for different population sizes.
For smaller populations, such as 4 million, we substitute into the regression equation to find:
\[\text{Predicted debt} = 19560.405 - 13.495 \times 4 = 19506.425\]
  • This prediction suggests that for a population of 4 million, the government debt per person would be approximately \(19506.425\) dollars.
Similarly, predictions can be made for larger populations, like 1367.5 million:
\[\text{Predicted debt} = 19560.405 - 13.495 \times 1367.5 = 1127.6925\]
  • This result indicates a predicted debt of \(1127.6925\) dollars per person for a very large population.
Predictive analysis thus provides insights based on current trends, but always remember, it relies heavily on the assumption that past patterns will continue into the future. Potential changes in societal or economic conditions might lead to different outcomes.

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Most popular questions from this chapter

Statistical studies show that a negative correlation exists between the number of flu cases reported each week throughout the year and the amount of ice cream sold in that particular week. Based on these findings, should physicians prescribe ice cream to patients who have colds and flu or could this conclusion be based on erroneous data and statistically unjustified? a. Discuss at least one lurking variable that could affect these results. b. Explain how multiple causes could affect whether an individual catches flu.

In 2015, an article published in the journal Breast Cancer Research and Treatment examined the impact of diabetes on the stages of breast cancer. The study concluded that diabetes is associated with advanced stages of breast cancer in patients and this could be a reason behind higher mortality rates. The researchers suggested looking at the possibility of race/ethnicity being a possible confounder. a. Explain what the last sentence means and how race/ ethnicity could potentially explain the association between diabetes and breast cancer. b. If race/ethnicity was not measured in the study and the researchers failed to consider its effects, could it be a confounding variable or a lurking variable? Explain the difference between a lurking variable and a confounding variable.

Most cars are fuel efficient when running at a steady speed of around 40 to \(50 \mathrm{mph}\). A scatterplot relating fuel consumption (measured in mpg) and steady driving speed (measured in mph) for a mid-sized car is shown below. The data are available in the Fuel file on the book's Web site. (Source: Berry, I. M. (2010). The Effects of Driving Style and Vehicle Performance on the Real-World Fuel Consumption of U.S. Light-Duty Vehicles. Masters thesis, Massachusetts Institute of Technology, Cambridge, MA.) a. The correlation equals \(0.106 .\) Comment on the use of the correlation coefficient as a measure for the association between fuel consumption and steady driving speed. b. Comment on the use of the regression equation as a tool for predicting fuel consumption from the velocity of the car. c. Over what subrange of steady driving speed might fitting a regression equation be appropriate? Why?

Example 9 related \(y=\) team scoring (per game) and \(x=\) team batting average for American League teams. For National League teams in 2010 , \(\hat{y}=-6.25+41.5 x\). (Data available on the book's website in the NL team statistics file.) a. The team batting averages fell between 0.242 and 0.272. Explain how to interpret the slope in context. b. The standard deviations were 0.00782 for team batting average and 0.3604 for team scoring. The correlation between these variables was 0.900 . Show how the correlation and slope of 41.5 relate in terms of these standard deviations. c. Software reports \(r^{2}=0.81 .\) Explain how to interpret this measure.

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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