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Chapter 3: Problem 27

You have data for many years on the average price of a barrel of oil and the average retail price of a gallon of unleaded regular gasoline. If you want to see how well the price of oil predicts the price of gas, then you should make a scatterplot with _____ as the explanatory variable. (a) the price of oil (b) the price of gas (c) the year (d) either oil price or gas price (e) time

Short Answer

Expert verified
(a) the price of oil

Step by step solution

01

Understand the Meaning of Explanatory Variable

In a scatterplot, the explanatory variable is used to predict or explain changes in the response variable. It's typically plotted on the horizontal axis (x-axis), whereas the response variable is plotted on the vertical axis (y-axis).
02

Identify the Variables

In this situation, you have two primary variables: the price of oil and the price of gas. You need to determine which of these will help predict the other.
03

Determine the Role of Each Variable

Read the problem statement carefully. You are interested in seeing how well the price of oil can predict the price of gas. This means the price of oil is the variable you use to predict, which makes it the explanatory variable.
04

Select the Correct Explanatory Variable

Based on the question's requirement to see how the price of oil predicts the price of gas, the price of oil is the explanatory variable. Therefore, in a scatterplot examining this relationship, the price of oil should be on the x-axis.

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Key Concepts

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

Understanding Scatterplots
A scatterplot is a graph that helps us visualize the relationship between two numerical variables. It is a simple yet powerful tool to identify patterns, trends, and potential correlations within data sets.
Scatterplots are plotted on a Cartesian coordinate grid, with one variable on the x-axis and the other on the y-axis. These axes allow us to see how the variables interact.
Typically:
  • The x-axis represents the explanatory variable.
  • The y-axis represents the response variable.
Each point on the scatterplot represents a pair of values for the two variables. Patterns in the scatterplot can indicate:
  • Positive correlation: As one variable increases, the other also tends to increase.
  • Negative correlation: As one variable increases, the other tends to decrease.
  • No correlation: No clear trend in how the variables relate.
Thus, scatterplots are essential tools in predictive analysis and determining relationships between different factors.
What is a Response Variable?
In the context of data analysis, the response variable is the outcome or dependent variable that you are trying to predict or explain.
It provides valuable insights into how other factors, known as explanatory variables, influence it.

For example, if you're studying how oil prices affect gas prices:
  • The gas price is the response variable because it's the outcome we want to understand.
  • The oil price is the factor that might explain changes in the gas price.
Keep in mind:
  • The response variable is typically plotted on the y-axis of a scatterplot.
  • It is dependent on the explanatory variable, meaning any change in the explanatory variable can potentially lead to a change in the response variable.
Understanding what qualifies as a response variable is crucial in setting up a correct analysis as it directs what you aim to predict.
The Role of Predictive Analysis
Predictive analysis is a branch of data analytics that uses historical data to forecast future outcomes. By examining existing patterns and trends, analysts can make informed predictions about future events.
In the context of oil and gas prices:
  • Predictive analysis would involve examining past interactions between oil and gas prices to predict how future changes in oil prices will affect gas prices.
  • A scatterplot can be a helpful tool in this analysis, as it visually demonstrates the relationship between oil prices (explanatory variable) and gas prices (response variable).
Predictive analysis goes beyond mere observation; it often involves sophisticated statistical models and algorithms to generate forecasts.
Additionally, such analysis requires:
  • Accurate data collection.
  • Proper identification of explanatory and response variables.
  • Careful interpretation to avoid spurious conclusions.
With the rise of big data, predictive analysis has become a critical resource in fields such as finance, marketing, and environmental science, among others.

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Most popular questions from this chapter

Measurements on young children in Mumbai, India, found this least-squares line for predicting height \(y\) from \(\operatorname{arm} \operatorname{span} x:\) $$\hat{y}=6.4+0.93 x$$ Measurements are in centimeters \((\mathrm{cm})\). In addition to the regression line, the report on the Mumbai measurements says that \(r^{2}=0.95\). This suggests that (a) although arm span and height are correlated, arm span does not predict height very accurately. (b) height increases by \(\sqrt{0.95}=0.97 \mathrm{~cm}\) for each additional centimeter of arm span. (c) \(95 \%\) of the relationship between height and arm span is accounted for by the regression line. (d) \(95 \%\) of the variation in height is accounted for by the regression line. (e) \(95 \%\) of the height measurements are accounted for by the regression line.

Exercise 10 (page 160 ) presented data on the lean body mass and resting metabolic rate for 12 women who were subjects in a study of dieting. Lean body mass, given in kilograms, is a person's weight leaving out all fat. Metabolic rate, in calories burned per 24 hours, is the rate at which the body consumes energy. Here are the data again. $$\begin{array}{llllllll}\hline \text { Mass: } & 36.1 & 54.6 & 48.5 & 42.0 & 50.6 & 42.0 & 40.3 & 33.1 & 42.4 & 34.5 & 51.1 & 41.2 \\\\\text { Rate: } & 995 & 1425 & 1396 & 1418 & 1502 & 1256 & 1189 & 9131124 & 1052 & 1347 & 1204 \\\\\hline\end{array}$$ (a) Use your calculator to help make a scatterplot. (b) Use your calculator's regression function to find the equation of the least-squares regression line. Add this line to your scatterplot from part (a). (c) Explain in words what the slope of the regression line tells us. (d) Calculate and interpret the residual for the woman who had a lean body mass of \(50.6 \mathrm{~kg}\) and a metabolic rate of 1502 .

What is the relationship between rushing yards and points scored in the 2011 National Football League? The table below gives the number of rushing yards and the number of points scored for each of the 16 games played by the 2011 Jacksonville Jaguars. $$\begin{array}{ccc}\hline \text { Game } & \text { Rushing yards } & \text { Points scored } \\\1 & 163 & 16 \\\2 & 112 & 3 \\\3 & 128 & 10 \\\4 & 104 & 10 \\\5 & 96 & 20 \\\6 & 133 & 13 \\\7 & 132 & 12 \\\8 & 84 & 14 \\\9 & 141 & 17 \\\10 & 108 & 10 \\\11 & 105 & 13 \\\12 & 129 & 14 \\\13 & 116 & 41 \\\14 & 116 & 14 \\ 15 & 113 & 17 \\\16 & 190 & 19 \\\\\hline\end{array}$$ (a) Make a scatterplot with rushing yards as the explanatory variable. Describe what you see. (b) The number of rushing yards in Game 16 is an outlier in the \(x\) direction. What effect do you think this game has on the correlation? On the equation of the leastsquares regression line? Calculate the correlation and equation of the least-squares regression line with and without this game to confirm your answers. (c) The number of points scored in Game 13 is an outlier in the \(y\) direction. What effect do you think this game has on the correlation? On the equation of the least-squares regression line? Calculate the correlation and equation of the least-squares regression line with and without this game to confirm your answers.

The gas mileage of an automobile first increases and then decreases as the speed increases. Suppose that this relationship is very regular, as shown by the following data on speed (miles per hour) and mileage (miles per gallon). $$\begin{array}{lccccc}\hline \text { Speed: } & 20 & 30 & 40 & 50 & 60 \\\\\text { Mileage: } & 24 & 28 & 30 & 28 & 24 \\\\\hline \end{array}$$ (a) Make a scatterplot to show the relationship between speed and mileage. (b) Calculate the correlation for these data by hand or using technology. (c) Explain why the correlation has the value found in part (b) even though there is a strong relationship between speed and mileage.

How sensitive to changes in water temperature are coral reefs? To find out, measure the growth of corals in aquariums where the water temperature is controlled at different levels. Growth is measured by weighing the coral before and after the experiment. What are the explanatory and response variables? Are they categorical or quantitative?
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