Chapter 15

• 15.1
  • (1) The correlation coefficient can be either positive or negative.
  • (2) The correlation coefficient always falls between − 1.00 and 1.00. (3) It is the strength, also called the magnitude, of the coefficient, not its sign, that indicates how large it is.
• 15.2 When two variables are correlated, there can be multiple explanations for that association. The first variable can cause the second variable; the second variable can cause the first variable; or a third variable can cause both the first and second variables. In fact, there may be more than one “third” variable causing both the first and second variables.
• 15.3
  • a. According to Cohen, this is a large (strong) correlation. Note that the sign (negative in this case) is not related to the assessment of strength.
  • b. This is just above a medium correlation.
  • c. This is lower than the guideline for a small correlation, 0.10.
• 15.4 Students will draw a variety of different scatterplots. The important thing to note is the closeness of data points to an imaginary line drawn through the data.
  • a. A scatterplot for a correlation coefficient of −0.60 might look like this:
    
    
  • b. A scatterplot for a correlation coefficient of 0.35 might look like this:
    
    
  • c. A scatterplot for a correlation coefficient of 0.04 might look like this:
    
    
• 15.5 It is possible that training while listening to music (A) causes an increase in a country’s average finishing time (B), perhaps because music decreases one’s focus on running. It is also possible that high average finishing times (B) cause an increase in the percentage of marathon runners in a country who train while listening to music (A), perhaps because slow runners tend to get bored and need music to get through their runs. It also is possible that a third variable, such as a country’s level of wealth (C), causes a higher percentage of runners who train while listening to music (because of the higher presence of technology in wealthy countries) (A) and also causes higher (slower) finishing times (perhaps because long-distance running is a less popular sport in wealthy countries with access to so many sport and entertainment options) (B). Without a true experiment, we cannot know the direction of causality.
• 15.6 The Pearson correlation coefficient is a statistic that quantifies a linear relation between two scale variables. Specifically, it describes the direction and strength of the relation between the variables.
• 15.7 The two issues are variability and sample size.
• 15.8
  
• 15.9 Step 1 is calculating the numerator:
  

  VARIABLE A (X)	(X − MX)	VARIABLE B (Y)	(Y − MY)	(X − MX)(Y − MY)
  8	1.812	14	3.187	5.775
  7	0.812	13	2.187	1.776
  6	−0.188	10	−0.813	0.153
  5	−1.188	9.5	−1.313	1.56
  4	−2.188	8	−2.813	6.155
  5.5	−0.688	9	−1.813	1.247
  6	−0.188	12	1.187	−0.223
  8	1.812	11	0.187	0.339
  MX = 6.118		MY = 10.813	Σ[(X − MX)(Y − MY)] = 16.782

  Step 2 is calculating the denominator:
  

  VARIABLE A (X)	(X − MX)	(X − MX)2	VARIABLE B (Y)	(Y − MY)	(Y − MY)2
  8	1.812	3.283	14	3.187	10.157
  7	0.812	0.659	13	2.187	4.783
  6	−0.188	0.035	10	−0.813	0.661
  5	−1.188	1.411	9.5	−1.313	1.724
  4	−2.188	4.787	8	−2.813	7.913
  5.5	−0.688	0.473	9	−1.813	3.287
  6	−0.188	0.035	12	1.187	1.409
  8	1.812	3.283	11	0.187	0.035
  	Σ(X − MX)2 = 13.966			Σ(Y − MY)2 = 29.969

  
  Step 3 is computing r :
  
• 15.10 Step 1: Population 1: Children like those we studied. Population 2: Children for whom there is no relation between observed and performed acts of aggression. The comparison distribution is made up of correlations based on samples of this size, 8 people, selected from the population.
  We do not know if the data were randomly selected, the first assumption, so we must be cautious when generalizing the findings. We also do not know if the underlying population distributions for witnessed aggression and performed acts of aggression by children are normally distributed. The sample size is too small to make any conclusions about this assumption, so we should proceed with caution. The third assumption, unique to correlation, is that the variability of one variable is equal across the levels of the other variable. Because we have such a small data set, it is difficult to evaluate this. However, we can see from the scatterplot that the data are somewhat consistently variable.
  Step 2: Null hypothesis: There is no correlation between the levels of witnessed and performed acts of aggression among children—H₀: ρ = 0. Research hypothesis: There is a correlation between the levels of witnessed and performed acts of aggression among children—H₁: ρ ≠ 0.
  Step 3: The comparison distribution is a distribution of Pearson correlations, r, with the following degrees of freedom: df_r = N − 2 = 8 − 2 = 6.
  Step 4: The critical values for an r distribution with 6 degrees of freedom for a two-tailed test with a p level of 0.05 are − 0.707 and 0.707.
  Step 6: The test statistic, r = 0.82, is larger in magnitude than the critical value of 0.707. We can reject the null hypothesis and conclude that a strong positive correlation exists between the number of witnessed acts of aggression and the number of acts of aggression performed by children.
• 15.11 Psychometricians calculate correlations when assessing the reliability and validity of measures and tests.
• 15.12 Coefficient alpha is a measure of reliability. To calculate coefficient alpha, we take, in essence, the average of all split-half correlations. That is, the items on a test are split in half and the correlation of those two halves is calculated. This process is done repeatedly for all possible “halves” of the test, and then the average of those correlations is obtained.
• 15.13
  • a. The test does not have sufficient reliability to be used as a diagnostic tool. As stated in the chapter, when using tests to diagnose or make statements about the ability of individuals, a reliability of at least 0.90 is necessary.
  • b. We do not have enough information to determine whether the test is valid. The coefficient alpha tells us only about the reliability of the test.
  • c. To assess the validity of the test, we would need the correlation between this measure and other measures of reading ability or between this measure and students’ grades in the nonremedial class (i.e., do students who perform very poorly in the nonremedial reading class also perform poorly on this measure?).
• 15.14 The large change in the correlation between college GPA and SAT scores means that a large part of the correlation between the two may actually be due to the relation between high school GPA and SAT scores. Once you account for that relation, the correlation between college GPA and SAT scores is greatly reduced.
• 15.15
  • a. The psychometrician could assess test–retest reliability by administering the quiz to 100 heterosexual female readers and then one week later readministering the test to the same 100 female readers. If their scores at the two times are highly correlated, the test would have high test–retest reliability. She also could calculate a coefficient alpha using computer software. The computer would essentially calculate correlations for every possible two groups of five items and then calculate the average of all of these split-half correlations.
  • b. The psychometrician could assess validity by choosing criteria that she believed assessed the underlying construct of interest, a boyfriend’s devotion to his girlfriend. There are many possible criteria. For example, she could correlate the amount of money each participant’s boyfriend spent on her last birthday with the number of minutes the participant spent on the phone with her boyfriend today or with the number of months the relationship ends up lasting.
  • c. Of course, we assume that these other measures actually assess the underlying construct of a boyfriend’s devotion, which may or may not be true! For example, the amount of money that the boyfriend spent on the participant’s last birthday might be a measure of his income, not his devotion.