Is there a correlation between studying and taking the exam?

Question

                       study      did not study           total
 took exam              35            17                   52
did not take the exam   12             5                   17
total                   47            22                   69

Is there a correlation between studying and taking the exam? Please provide me with a hint at least. This problems is very different from any other I have done.

Answer 1

(1) In statistics and probability, the word 'correlation' has a very specific technical meaning, so you should avoid using it for just any kind of 'association'.

(2) You are asking whether your data show evidence that in the population from which you sampled, the events $A=\{\text{studied}\}$ and $B = \{\text{took exam}\}$ are associated (dependent).

(3) With this hint about the proper terminology, you can look for a 'chi-squared test of independence.' For this simple '2-by-2' case, there are many different versions of the formula, so I won't attempt to illustrate the computations.

(4) For your data, there is essentially no evidence of dependence. For example, the overall proportion who studied is $\hat P\{\text{studied}\} = 47/69 \approx 0.681$ and the proportion taking the exam is $\hat P\{\text{took exam}\} = 52/69 \approx 0.754$.

Intuitively then, if these events were independent we would have $\hat P\{\text{both}\} = 0.681(0.754) \approx 0.51.$ However, directly from your data the observed proportion who both studied and took the exam is $35/69 \approx 0.51.$ So your data are clearly (and astonishingly) consistent with independence of the two events.

(5) Speculative: Below is output from one kind of statistical software for such a chi-squared test. Perhaps you can figure out how to do these computation from what is in your text, or perhaps your text has a different formulation of this test.

 Chi-Squared Test:

 Expected counts are printed below observed counts
 Chi-Squared contributions are printed below expected counts

     Exam\Study     Yes     No  Total
     Yes             35     17     52
                  35.42  16.58
                  0.005  0.011

     No              12      5     17
                  11.58   5.42
                  0.015  0.033

     Total           47     22     69

 Chi-Sq = 0.063, DF = 1, P-Value = 0.801

The Chi-squared statistic (0.063) would need to exceed 3.84 before you could claim an association between studying and taking the exam. (Equivalently, the P-value would need to be below 0.05.)