For the complete answers see the final slides of the week 6 slide show
Hypothesis: Rank causes cynicism
Zero-order table
We do not need to convert these cell frequencies into percentages. Why? When selecting the sample, I stratified (as usual) by the independent variable. By choosing 100 officers and 100 supervisors, the distributions along the dependent variable, at each value of the independent variable, are automatically percentages (PER 100).
Is there are relationship? You bet. Note that as we "switch" the independent variable RANK from officers to supervisors, the distribution of cases along the dependent variable changes, from 20 LOW and 80 HIGH to 50/50. Changes in the independent variable go along with changes in the dependent variable. So there is an association between the variables.
First-order partial tables: "replication"
Here we converted frequencies to percentages because the totals at each level of the independent variable are no longer 100. Inspecting the percent tables for both levels (male, female) of the control variable GENDER, we notice that each depicts a relationship between RANK and CYNICISM that resembles the relationship in the zero-order table. We have "replicated" our original finding. GENDER, whether male or female, does not tell us anything new. It does not change our original opinion - changes in rank appear to cause changes in cynicism. BUT suppose the first-order tables came out looking like this:
First-order partial tables: "specification"
Although there still seems to be a relationship between rank and cynicism for males, there does not seem to be a relationship between rank and cynicism for females. For females, when we "switch" the independent variable from officers to supervisors, the distributions of cases along dependent variable CYNICISM remain the same. There seems to be no connection - no relationship - between rank and cynicism.
When some values of a control variable are consistent with the zero-order relationship, but others are not, we call the effect of the control variable "specification".
First-order partial tables: "explanation"
Above, there seems to be no relationship between rank and cynicism for females. IF there had been no relationship for males, then control variable GENDER would have completely "explained away" the relationship in the zero-order table. We could then say that rank does not cause cynicism - gender does!
When an association between variables in a zero-order table is rejected at every level of a control variable, we say that the control variable "explains away" the zero-order relationship.
The scattergram depicts a very strong positive relationship between variables. Estimated r is + .80 or + .90
Below are the same variables rescaled as categorical. Note that the distribution of the dependent variable runs vertically at each value of the independent variable.
At the Short value of the independent variable (height), the distribution of the dependent variable (weight) is skewed completely to Low. But when we change the value of the independent variable to Tall, the distribution of the dependent variable flips, with most cases now High.
Adjusting the value of the independent variable does change the value of the dependent variable. Considering the magnitude of the change, their association seems strong even after they are rescaled from continuous to categorical.
HOMEWORK ANSWERS - WEEKS 11 & 12
Standard deviation for sample 1: .99
Standard deviation for sample 2: .97
Standard error of the mean based on sample 1: .33
Standard error of the mean based on sample 2: .32
95% Confidence interval into which the population mean should fall, based on sample 1:
Left limit = 2.25 Right limit = 3.55
95% Confidence interval into which the population mean should fall, based on sample 2:
Left limit = 1.77 Right limit = 3.03
HOMEWORK ANSWERS - WEEKS 13 -16
Pooled sample variance = .96
Standard error of the difference between means= .44
t- test = 1.14
df = 18
TWO-tailed test (we did not predict which sample means would be larger)
We CANNOT reject the null hypothesis. The probability that the difference between means is due to chance exceeds 5 in 100.
