In recent years, the use of e-cigarettes, or vaping, has become the subject of considerable public interest. Though e-cigarettes are marketed to help smokers kick the habit, many non-smokers--particularly minors--now vape. A recent study published in the journal Pediatrics selected a random sample of 2084 11^th and 12^th grade students in California (Barrington-Trimos, et al., 2015).[i]The sample data are found in a contingency table with current/past e-cigarette use (User vs. NonUser) by row and gender (Female vs. Male) by column.

1. From the sample data, compute the marginal frequencies and then the marginal probabilities that a student in the dataset (i) is a male, (ii) has used e-cigarettes.
2. Now compute the conditional probability that a student has used e-cigarettes, if that student is a male. Also from the sample data, compute the conditional probability that a student has used e-cigarettes, if that student is female; i.e., not male (hint: you will need to use the identity that relates marginal, conditional, and joint probabilities).
3. From the sample data, compute the expected frequencies (assuming independence) of e-cigarette use (User vs. NonUser) and gender (Female vs. Male).
4. Use the expected and actual frequencies to determine whether e-cigarette usage and gender are independent. Specifically, report the appropriate test statistic, the p-value for that teststatistic and your conclusion, based on that p-value.