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Chapter 3: Problem 1

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

Short Answer

Expert verified
a. Carat (explanatory), price (response); b. Dosage (explanatory), severity (response); c. Construction type (explanatory), top speed (response); d. Type of college (explanatory), graduation rate (response).

Step by step solution

01

Understand Response and Explanatory Variables

In any given scenario, the response variable (also known as the dependent variable) is the outcome we are interested in observing or measuring. The explanatory variable (or independent variable) is what we manipulate or what we believe is causing changes in the response variable. It's important to determine which variable is most likely influencing the other.
02

Analyze Each Pair of Variables

For each given pair, consider which variable might be influencing the other. (a) Carat and price of a diamond: Here, the carat (weight) of a diamond typically influences its price. So, the carat weight is the explanatory variable, and the price is the response variable. (b) Dosage and severity of adverse events: In this scenario, dosage is expected to impact the severity of adverse events. Hence, dosage is the explanatory variable, while severity is the response variable.
03

Continue Analyzing Remaining Pairs

Continue examining each pair to identify the explanatory and response variables. (c) Top speed and construction type of a roller coaster: It is likely that the construction type (wood or steel) influences the roller coaster's top speed. Thus, construction type is the explanatory variable, and top speed is the response variable. (d) Type of college and graduation rate: The type of college (private or public) is more likely to affect the graduation rate. Therefore, the type of college is the explanatory variable, and graduation rate is the response variable.

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Key Concepts

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

Dependent Variable
In the world of data analysis, the dependent variable plays a crucial role. It is also known as the response variable. This variable is what researchers measure in an experiment. It reflects the outcomes or results based on changes in other variables. Think of it like the child in a parent-child relationship. It reacts to what the parent does. Here are some key points to remember about dependent variables:
  • The dependent variable is what you're trying to understand or predict.
  • It "depends" on the independent variable or variables.
  • In scientific experiments, it is the outcome you observe after manipulation of other factors.
In the context of our exercise, the price of a diamond and the graduation rate are examples of dependent variables, as they change based on carat weight and type of college, respectively.
Independent Variable
The independent variable is the one that you manipulate or change to observe its effects. It is also called the explanatory variable. Think of it as the "cause" to the "effect" of a dependent variable. Some characteristics of independent variables include:
  • They are presumed to influence or determine the dependent variable.
  • They are the "input" variables that get altered to see how they impact the dependent variable.
  • In experiments, they are manipulated to observe changes in other variables.
In our exercise examples, carat weight is an independent variable affecting the price, just like the dosage impacts the severity of adverse events. These independent variables explain why a change occurs in the dependent variables.
Causal Relationships
A causal relationship describes the connection between two variables where a change in one directly influences the other. This is a crucial aspect of understanding dependent and independent variables. Identifying causal relationships is important because:
  • It helps to establish if and how one variable truly impacts another.
  • They allow for predictions of outcomes if one variable changes.
  • In research, establishing causality can guide focus and resource allocation for interventions.
When you are considering variables like dosage and severity of adverse events, determining a causal relationship helps to understand if and how increasing the dosage escalates the event severity. However, it鈥檚 important to note that not all relationships are causal, and some might just be correlational due to confounding factors.
Statistical Analysis
Statistical analysis is a method used to collect, review, and interpret data to discover relationships between variables. It is essential for testing hypotheses about dependent and independent variables. Some benefits of statistical analysis are:
  • It allows for the quantification of relationships between variables.
  • Statistical tests help determine the significance and strength of these relationships.
  • It provides a structured way to make inferences from data samples to larger populations.
In the context of our examples, statistical analysis could quantify the impact of carat weight on the price of a diamond or validate the effect of college type on graduation rates. Without statistical analysis, discerning meaningful patterns or causal links from raw data would be much more challenging.

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Most popular questions from this chapter

Wage bill of Premier League Clubs Data of the Premier League Clubs' wage bills was obtained from www.tsmplug .com. For the response variable \(y=\) wage bill in millions of pounds in 2014 and the explanatory variable \(x=\) wage bill in millions of pounds in \(2013, \hat{y}=-1.537+1.056 x\). a. How much do you predict the value of a club's wage bill to be in 2014 if in 2013 the club had a wage bill of (i) \(拢 100\) million, (ii) \(拢 200\) million? b. Using the results in part a, explain how to interpret the slope. c. Is the correlation between these variables positive or negative? Why? d. A Premier League club had a wage bill of \(拢 100\) million in 2013 and \(拢 105\) million in \(2014 .\) Find the residual and interpret it.

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 \(r=-0.28\).

For the 100 cars on the lot of a used-car dealership, would you expect a positive association, negative association, or no association between each of the following pairs of variables? Explain why. a. The age of the car and the number of miles on the odometer b. The age of the car and the resale value c. The age of the car and the total amount that has been spent on repairs d. The weight of the car and the number of miles it travels on a gallon of gas e. The weight of the car and the number of liters it uses per \(100 \mathrm{~km}\).

How much do seat belts help? In \(2013,\) data was collected from the U.S. Department of Transportation and the Insurance Institute for Highway Safety. According to the collected data, the number of deaths per 100,000 individuals in the U.S would decrease by 24.45 for every 1 percentage point gain in seat belt usage. Let \(\hat{y}=\) predicted number of deaths per 100,000 individuals in 2013 and \(x=\) seat belt use rate in a given state. a. Report the slope \(b\) for the equation \(\hat{y}=a+b x\). b. If the \(y\) intercept equals \(32.42,\) then predict the number of deaths per 100,000 people in a state if (i) no one wears seat belts, (ii) \(74 \%\) of people wear seat belts (the value for Montana), (iii) \(100 \%\) of people wear seat belts.

NAEP scores In 2015 , eighth-grade math scores on the National Assessment of Educational Progress had a mean of 283.56 in Maryland compared to a mean of 284.37 in Connecticut (Source: http://nces.ed.gov/nationsreportcard/ naepdata/dataset.aspx). a. Identify the response variable and the explanatory variable. b. The means in Maryland were respectively \(274,284,285,\) 291 and 294 for people who reported the number of pages read in school and for homework, respectively as \(0-5,6-10,11-15,15-20\) and 20 or more. These means were 270,281,284,289 and 293 in Connecticut. Identify the third variable given here. Explain how it is possible for Maryland to have the higher mean for each class, yet for Connecticut to have the higher mean when the data are combined. (This is a case of Simpson's paradox for a quantitative response.)
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