Background

Students in Math 325: Intermediate Statistics anonymously answered a survey to indicate how they feel about certain aspects of the course. They rate aspects on a scale from 0-3, 0 indicating they feel they have lost learning from an aspect and 3 indicating they feel they have learned from the aspect. We will look at these rating as Unsatisfied (0 to 1), Satisfied (1 to 2), and Pleased (2 to 3). The two aspects we will analyze are Critiquing others’ work in the course and Being Critiqued by others in the course.

Hypothesis

Our question for our data is: Do students who like/dislike critiquing have the same feelings towards being critiqied? In other words, are the satisfaction levels of critiquing other students’ work and being critiqued by other students associated? Our null and alternative hypothesis are as follows:

\[ H_0: \text{The Critiquing satisfaction and Being Critiqued satisfaction are independent}. \] \[ H_a: \text{The Critiquing satisfaction and Being Critiqued satisfaction are associated (not independent)}. \]

We will be using a level of significance \(\alpha=0.05\).

Analysis

First, let’s look at a table of counts for Critquing (rows) and Being Critiqued (columns). We do see that the some of the smallest counts are for being pleased with one thing and unsatisfied with the other. If students are pleased with Being Critiqued, they are not likely to be unstatisfied with Critiquing with a count of 3. Also, some of the largest counts (84 and 82) are matched on levels of satisfaction for Critiquing and Being Critiqued. So, we should expect to have some association with Critiquing and Being Critiqued.

qs2<-qs
qs2$Critiquing<-cut(qs2$Critiquing, c(-1,1.5,2.5,3),c("Unsatisfied","Satisfied","Pleased"),ordered_result = TRUE)
qs2$BeingCritiqued<-cut(qs2$BeingCritiqued, c(-1,1.5,2.5,3),c("Unsatisfied","Satisfied","Pleased"),ordered_result = TRUE)
survtable<-table(qs2$Critiquing,qs2$BeingCritiqued)
pander(survtable)
  Unsatisfied Satisfied Pleased
Unsatisfied 25 12 3
Satisfied 30 84 46
Pleased 18 53 82

We see our barplot shows some of our conjectures are strong. However, we must run a test to get a \(p\)-value to determine if our conjectures are correct.

barplot(survtable, beside=TRUE, col=c("springgreen","springgreen2","springgreen3"),legend.text=TRUE,args.legend = list(x="topleft",bty="n",title="Critiquing"),xlab="Being Critiqued",main="Student Satisfaction \n Math 325: Intermediate Statistics")

Now, we will run a Chi-Squared Test to determine if we reject our null hypothesis and conclude that the satisfaction levels of Critiquing and Being Critiqued are associated. We will also look at our expected counts to determine if our Chi-Squared Test requirements are met.

qs2Chi<-chisq.test(survtable)
pander(qs2Chi)
Pearson’s Chi-squared test: survtable
Test statistic df P value
71.17 4 1.287e-14 * * * 
pander(qs2Chi$expected)
  Unsatisfied Satisfied Pleased
Unsatisfied 8.272 16.88 14.84
Satisfied 33.09 67.54 59.38
Pleased 31.64 64.58 56.78

All expected counts are greater than 5, so the requirements are met.

Our test results show a \(p\)-value of \(p=1.287e-14\), which is much smaller than our level of significance \(\alpha=0.05\), so we reject our null hypothesis and conclude that the satisfaction levels of Critiquing and Being Critiqued are associated.

Let us look at our residuals to see how much our observed counts differ from the expected counts if our null hypothesis were true.

pander(qs2Chi$residuals)
  Unsatisfied Satisfied Pleased
Unsatisfied 5.816 -1.189 -3.074
Satisfied -0.5368 2.003 -1.736
Pleased -2.425 -1.441 3.347

We see the greatest difference between what was observed and what would be expected is where students are unsatisfied with both aspects of Critiquing and Being Critiqued, with a difference of 5.816 more counts. This is probably what contributed most to our test statistic and resulting \(p\)-value.

Interpretation

We can interpret how our alternative hypothesis applys to the students in Math 325. We see that students who are satisfied/unsatisfied with one aspect (Critiquing and Being Critiqued) will most likely feel the same way about the other aspect. Maybe this is because the students who are unsatisfied find both aspects of the course to be time consuming and unaffective. Those who are satisfied, and likely satisfied with both, may find these aspects helpful and a learning experience that is worth the time. Another survey could be conducted to see why students are satisfied/unsatisfied, but for now, we can see that they will have similar satisfaction levels for Critiquing others’ work and Being Critiqued by others.