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Chapter 4: Problem 40

Sloppy Writing about Correlation. Fach of the following statements contains a blunder. Explain in each case what is wrong- a. "There is a high carrelation between the gender of an adult and and their political affiliation. b. "We found a strong negative correlation \(\\{r=-1.09\\}\) between the amount of time spent on social media and the number of boolss read in the last year. c. "The carrelation hetween height and weight of the subjects was \(r=0.63\) centimeter per kilogram."

Short Answer

Expert verified
a. Correlation applies to quantitative data only; use a Chi-Square test. b. Correlation coefficient must be between -1 and 1; -1.09 is invalid. c. Correlation is unitless; it shouldn't have units like cm/kg.

Step by step solution

01

Analyze Statement a

The statement claims there is a high correlation between gender and political affiliation. The error here is the application of correlation, which is a statistical measure that requires quantitative data. Gender and political affiliation are qualitative, categorical variables. As such, correlation is not appropriate for such data. Instead, the researcher might consider measures like the Chi-Square test for independence to explore the relationship.
02

Analyze Statement b

The statement mentions a correlation value of \( r = -1.09 \). The error in this statement is that correlation coefficients must fall within the range of -1 to +1. A value of -1.09 is invalid. A correct correlation coefficient would imply a perfect negative correlation if it were exactly -1, but anything beyond this range is incorrect.
03

Analyze Statement c

The error in this statement is related to the units attached to the correlation coefficient. Correlation is a dimensionless measure and should not have any units like centimeters per kilogram. The value \( r = 0.63 \) already appropriately indicates the strength of the linear relationship without any units.

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Key Concepts

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

Quantitative Data
Quantitative data is all about numbers and measurements. When we talk about quantitative data, we are referring to data that can be quantified and assigned a numerical value. Numbers like height, weight, and time spent on activities are examples of quantitative data. They allow us to perform arithmetic operations and statistical measurements like mean, median, variance, and standard deviation.

Quantitative data can be divided into two types:
  • Discrete Data: These are countable in a finite amount of time, like the number of books read in a year.
  • Continuous Data: These can be measured and divided into fractions, such as height or weight.
This type of data is essential when applying statistical techniques like correlation analysis since these methods require numerical inputs to calculate meaningful outputs.
Qualitative Variables
Qualitative variables, also known as categorical variables, refer to data that can be categorized based on characteristics or attributes. These variables are non-numerical and describe qualities. Examples include gender, political affiliation, and color categories.

With qualitative data, you can't perform typical mathematical operations. Instead, categorization and classification are essential. Some analysis methods that can handle qualitative variables include:
  • Chi-Square Test: Examines the independence of qualitative variables by comparing observed counts against expected counts.
  • Mode and Frequency: Useful in identifying the most common category in a dataset.
In the context of correlation analysis, it's crucial not to use correlation coefficients directly with qualitative variables. Tools like the chi-square test are better suited.
Correlation Coefficient
The correlation coefficient is a statistical measure that describes the extent to which two quantitative variables are linearly related. It ranges between -1 and +1, where:
  • 1: Perfect positive linear relationship.
  • 0: No linear relationship.
  • -1: Perfect negative linear relationship.
A common formula for calculating the correlation coefficient is Pearson's formula, which considers both the variance of each variable and the covariance between them.

It is crucial to note:
  • The coefficient must remain between -1 and +1.
  • No units should be attached as it is a dimensionless measure.
These constraints ensure the correlation coefficient provides a meaningful measure of linear relationship strength and direction.
Chi-Square Test
The Chi-Square Test is a statistical method used to examine the relationships between categorical variables. This test checks whether the distributions of observed frequencies match expected frequencies under the assumption of independence.

Key features of the Chi-Square Test include:
  • Used for qualitative data, especially when testing for independence.
  • Calculated as the sum of squared differences between observed and expected frequencies, divided by the expected frequencies.
  • Works best with larger sample sizes to satisfy its approximation assumptions.
Use the chi-square test when dealing with variables like gender and political preference, where correlation coefficients do not apply. This way, you can better understand the possible dependences between qualitative variables without resorting to inappropriate statistical measures like correlation.

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Most popular questions from this chapter

Does Fast Driving Waste Fuel? How does the fuel consumption of a car change as its speed increases? Here are data for a 2013 Volkswagen Jetta Diesel. Speed is measured in miles per hour, and fuel consumption is measured in miles per gallon: \(:-6\). FASTDF \begin{tabular}{|l|l|l|l|l|l|l|l|} \hline Speed & 20 & 30 & 40 & 50 & 60 & 70 & 30 \\ \hline Fuel & \(49.0\) & \(67.9\) & \(66.5\) & 59 & \(50.4\) & \(44.8\) & \(39.1\) \\ \hline \end{tabular} a. Make a scatterplot. (Which is the explanatory variable?) b. Describe the form of the relat ionship. It is not linear. Explain why the form of the relationship makes sense. c. It does not make sense to describe the variables as either positively associated or negatively associated. Why? d. Is the relationship reasonably strong or quite weak? Explain your answer.

Researchers asked mothers how much soda (in ounces) their kids drank in a typical day. They also asked these mothers to rate how aggressive their kids were on a scale of 1 to 10 , with larger values corresponding to a greater degree of aggression. 12 The correlation between amount of soda consumed and aggression rating was found to be \(r=0.3\). If the researchers had measured amount of soda consumed in liters instead of ounces, what would be the correlation? (There are 35 ounces in a liter.) a. \(0.3 / 35=0.009\) b. \(0.3\) c. \((0.3)(35)=10.5\)

Changing the Units. The sea surface temperatures in Exercise 4.10 are measured in degrees Celsius and growth in centimeters per year. The correlation between sea surface temperature and coral growth is \(r=-0.8111\). If the measurements were made in degrees Fahrenheit and inches per year, would the correlation change? Explain your answer.

Researchers measured the percentage body fat and the preferred amount of salt (percent weight/volume) for several children. Here are data for seven children: \(: 13\) SALT \begin{tabular}{|l|l|l|l|l|l|l|l|l|l|} \hline Preferred amount of salt \(x\) & \(0.2\) & \(0.3\) & \(0.4\) & \(0.5\) & \(0.6\) & \(0.8\) & \(1.1\) & \\ \hline Percentage body fat \(y\) & 20 & 30 & 22 & 30 & 38 & 23 & 30 & \\ \hline \end{tabular} Use your calculator or software: The correlation between percentage body fat and preferred amount of salt is about a. \(r=0.08\). b. \(r=0.3\). C. \(r=0.8\).

What are all the values that a correlation \(r\) can possibly take? a. \(r \geq 0\) b. \(0 \leq r \leq 1\) c. \(-1 \leq r \leq 1\)
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