Negative Correlation: Definition, Examples, and How to Interpret

Problem 6.7 

(a) Estimate whether the following pairs of scores for X and Y reflect a positive relationship, a negative relationship, or no relationship. Hint: Note any tendency for pairs of X and Y scores to occupy similar or dissimilar relative locations. 

Solution: There is a negative correlation between X and Y as the scatter plot reveals an increase in the X values for every corresponding decrease in the Y values. 

(b) Construct a scatter plot for X and Y. Verify that the scatter plot does not describe a pronounced curvilinear trend. 

Solution: 

Since the points of the plot produce an upper-left to lower right pattern, signifying a negative correlation between variables X and Y, there is a lack of pronounced curvilinearity in the figure. 

(c) Calculate r using the computation formula (6.1) 

Solution: 

	X	Y
	64	66
	40	79
	30	98
	71	65
	55	76
	31	83
	61	68
	42	80
Sum	57  451	72  687

Problem 6.10 On the basis of an extensive survey, the California Department of Education reported an r of -.32 for the relationship between the amount of time spent watching TV and the achievement test scores of children. Each of the following statements represents a possible interpretation of this finding. Indicate whether each is True or False. 

Every child who watches a lot of TV will perform poorly on the achievement tests. (False) 

Extensive TV viewing causes a decline in test scores. (False) 

Children who watch little TV will tend to perform well on the tests. (True) 

Children who perform well on the tests will tend to watch little TV. (True) 

If Gretchen’s TV-viewing time is reduced by one-half, we can expect a substantial improvement in her test scores. (False) 

TV viewing could not possibly cause a decline in test scores. (False) 

Problem 6.11 Assume that an r of .80 describes the relationship between daily food intake, measured in ounces, and body weight, measured in pounds, for a group of adults. Would a shift in the units of measurements from ounces to grams and from pounds to kilograms change the value of r? Justify your answer. 

Solution: As correlation r has no units, a shift in the units of measurement from ounces to grams and from pounds to kilograms will have no effect on its value. 

Chapter 7 

Problem 7.8 Each of the following pairs represents the number of licensed drivers (X) and the number of cars (Y) for seven houses in my neighborhood: 

	DRIVERS  (X)	CARS (Y)
	5	4
	5	3
	2	2
	2	2
	3	2
	1	1
	2	2
Sum	20	16

Construct a scatterplot to verify a lack of pronounced curvilinearity. 

Solution: 

Since the points on the line represent a positive correlation between variables X and Y, it signifies the absence of a pronounced curvilinearity. 

Determine the least squares equation for these data. (Remember, you will first have to calculate r, SS y and SS x ) 

Solution: 

Least squares equation = 

Determine the standard error of estimate, S y│x , given that n = 7. 

Solution: 

Predict the number of cars for each of the two new families with two and five drivers. 

Solution: 

 5 

Problem 7.10 Assume that r 2 equals .50 for the relationship between height and weight for adults. Indicate whether the following statements are true or false. 

Fifty percent of the variability in heights can be explained by variability in weights. (True) 

There is a cause-effect relationship between height and weight. (False) 

The height of 50% of adults can be predicted exactly from their weights. (False) 

50 % of the variability in weights is predictable from heights. (True) 

Problem 7.13 In the original study of regression towards the mean, Sir Francis Galton noticed a tendency for offspring of both tall and short parents to drift towards the mean height for offspring and referred to this tendency as “regression towards mediocrity.” What is wrong with the conclusion that eventually all heights will be close to their mean? 

Answer: Sir Francis Galton’s conclusion, which implies that eventually all heights move closer to their mean over generations, is incorrect since a child inherits its genetic makeup solely from its parents and the gene responsible for expressing height in individuals may be either a dominant or a recessive trait. For example, a gene for tall height in the offspring may or may not be expressed depending upon the environmental factors as well as the sequence of genes it receives exclusively from its parents (whether it is a homozygous or a heterozygous gene pair). Hence, the phenomenon of regression towards mediocrity is inadequate in explaining height mutations across generations of offspring as both the expression of genes and various environmental factors are actually the factors responsible for the science of gene inheritance.