11. Correlation

Given a dataset where as one variable increases, the other variable tends to decrease moderately, which of the following values is the most likely correlation coefficient for this data?

Which value of the correlation coefficient $r$ indicates a stronger linear relationship: $r=0.85$ or $r=-0.65$?

Testing for a Linear Correlation

In Exercises 13–28, construct a scatterplot, and find the value of the linear correlation coefficient r. Also find the P-value or the critical values of r from Table A-6. Use a significance level of α = 0.05. Determine whether there is sufficient evidence to support a claim of a linear correlation between the two variables. (Save your work because the same data sets will be used in Section 10-2 exercises.)

Taxis Using the data from Exercise 15, is there sufficient evidence to support the claim that there is a linear correlation between the distance of the ride and the tip amount? Does it appear that riders base their tips on the distance of the ride?

What would you conclude about a set of quantitative bivariate data with a linear correlation coefficient of -1? How would the scatterplot for such data appear?

A marketing researcher analyzed advertising budget vs. monthly sales revenue for small retail stores and found that typically the stores that spent more on advertising saw higher sales revenues. However, the relationship wasn't perfect - some stores advertised more but saw fewer sales due to poor location, customer preferences, or bad timing. Which of the following is the most likely value for the correlation coefficient $r$ between advertising budget and sales revenue?

[DATA] Putting It Together: Exam Scores The data below represent scores earned by students in Sullivan’s Elementary Algebra course for Chapter 2 (Linear Equations and Inequalities in One Variable) and Chapter 3 (Linear Equations and Inequalities in Two Variables). Completely summarize the relation between Chapter 2 and Chapter 3 exam scores, treating Chapter 2 exam scores as the explanatory variable. Write a report detailing the results of the analysis including the presence of any outliers or influential points. What does the relationship say about the role Chapter 2 plays in a student’s understanding of Chapter 3?

What is the most likely value of the correlation coefficient between a stock-index mutual fund and the $S\&P500$ index?

If the coefficient of determination $r^2$ is high, can we conclude that there is a causal relationship between the two variables? Choose the best answer.

Suppose you are shown four scatterplots, each with a calculated correlation coefficient. Which of the following correlation coefficients indicates the strongest linear relationship between the two variables?

A negative correlation means $\dots$.

[DATA] American Black Bears The American black bear (Ursus americanus) is one of eight bear species in the world. It is the smallest North American bear and the most common bear species on the planet. In 1969, Dr. Michael R. Pelton of the University of Tennessee initiated a long-term study of the population in the Great Smoky Mountains National Park. One aspect of the study was to develop a model that could be used to predict a bear’s weight (since it is not practical to weigh bears in the field). One variable thought to be related to weight is the length of the bear. The following data represent the lengths and weights of 12 American black bears.

c. Determine the linear correlation coefficient between weight and length.

d. Does a linear relation exist between the weight of the bear and its length?

1. Two variables have a positive linear correlation. Does the dependent variable increase or decrease as the independent variable increases? What if the variables have a negative linear correlation?

Which of the following is a property of the linear correlation coefficient $r$?

A correlation coefficient is a statistic that tells the

What are some advantages of the Spearman rank correlation coefficient over the Pearson correlation coefficient?