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Chapter 12: Problem 13

Although the slope does not measure association, it is useful for comparing effects for two variables that have the same units. For the Internet Use data file of 33 nations on the text \(\mathrm{CD},\) let \(x=\mathrm{GDP}\) (thousands of dollars per capita). For predicting \(y=\) percentage Internet penetration (the percentage of adult users), the prediction equation is \(\hat{y}=0.1239+0.0157 x\). For predicting \(y=\) percentage Facebook penetration, the prediction equation is \(\hat{y}=0.081+0.0075 x\) a. Explain how to interpret the two slopes. b. For these nations, explain why a one-unit increase in GDP has a slightly greater impact on the percentage using Facebook than on the percentage using the Internet.

Short Answer

Expert verified
Internet penetration is more sensitive to GDP changes than Facebook penetration.

Step by step solution

01

Understanding the Slope for Internet Penetration

The slope for the Internet penetration prediction equation is 0.0157. This means that for every one-unit increase in GDP (i.e., an increase of $1000 per capita), the Internet penetration percentage increases by 0.0157. In other words, there is a positive relationship between GDP and Internet usage percentage, indicating that wealthier countries tend to have higher Internet penetration.
02

Understanding the Slope for Facebook Penetration

The slope for the Facebook penetration prediction equation is 0.0075. This means that for every additional thousand dollars in GDP, the Facebook penetration percentage increases by 0.0075. Similar to Internet penetration, there is a positive relationship between GDP and Facebook usage percentage, suggesting that as GDP increases, Facebook penetration also tends to rise.
03

Comparing Slopes to Assess Impact

To compare impacts, examine the magnitudes of the two slopes: 0.0157 for Internet use and 0.0075 for Facebook use. A one-unit increase in GDP has a greater impact on Internet penetration than Facebook penetration because 0.0157 is greater than 0.0075. Thus, even though both are positively correlated with GDP, Internet penetration increases more significantly with GDP increases than Facebook penetration does.

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Key Concepts

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

Slope Interpretation
Slope interpretation in linear regression is like deciphering the effect of one unit increase in the independent variable (x) on the dependent variable (y). In the context of this exercise, we have two slopes from distinct prediction equations involving GDP (\(x\)) as the independent variable: one for predicting Internet penetration and another for Facebook penetration.
The Internet penetration slope of 0.0157 indicates that for each $1000 increase in GDP per capita, the Internet usage percentage is expected to rise by 0.0157. Similarly, the Facebook penetration slope is 0.0075, meaning the percentage of Facebook users would grow by 0.0075 for the same GDP increase. Thus:
  • A slope tells us how much the dependent variable is expected to increase or decrease.
  • A positive slope, as seen here, implies a direct proportionate rise with GDP.
Remember, slopes provide insight into direct changes but do not account for other potential influencing factors or causal relationships.
Predictive Modeling
Predictive modeling is a method used to forecast outcomes based on input data. It is a significant aspect in fields ranging from business to health to economy. In this exercise, predictive models are crafted using linear regression equations to make forecasts regarding Internet and Facebook penetration.
Linear regression predicts by establishing a relationship between an independent variable (like GDP) and a dependent variable (such as Internet usage). The equation follows the format \(\hat{y} = a + bx\), where:
  • \(\hat{y}\) is the predicted value of the dependent variable.
  • \(a\) is the intercept, representing the expected value of \(y\) when \(x = 0\).
  • \(b\) is the slope, delineating the expected change in \(y\) for a one-unit increase in \(x\).
These models are not just mathematical functions but powerful tools to extrapolate potential future trends from present data. They help stakeholders make informed decisions based on current metrics.
Association Measurement
While interpreting slopes is vital for understanding specific impacts, it's crucial to differentiate them from association measures. Association measurement in statistics involves understanding how variables relate to one another, typically quantified by correlation coefficients.
In the given problem, although the slope can infer directionality (how \(x\) influences \(y\)), it does not tell us about the strength or reliability of this relationship. That's where correlation plays a role. Here are key differences:
  • Slope estimates the change rate of one variable in response to another.
  • Correlation quantifies the degree and direction of association between two variables.
Remember, a strong slope does not imply a perfect link; it merely indicates a trend. When analyzing data, a complete picture emerges from considering slopes and correlation together. Association measurement ensures we comprehend how closely and significantly variables interconnect.

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Most popular questions from this chapter

The table shows the approximate U.S. population size (in millions) at 10 -year intervals beginning in 1900 . Let \(x\) denote the number of decades since 1900 . That is, 1900 is \(x=0,1910\) is \(x=1,\) and so forth. The exponential regression model fitted to \(y=\) population size and \(x\) gives \(\hat{y}=81.14 \times 1.1339^{x}\) a. Show that the predicted population sizes are 81.14 million in 1900 and 323.3 million in 2010 . b. Explain how to interpret the value 1.1339 in the prediction equation. c. The correlation equals 0.98 between the log of the population size and the year number. What does this suggest about whether or not the exponential regression model is appropriate for these data?

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