In this case study we will be using the smallpox dataset. The smallpox dataset looks at 6,224 individuals that were in Boston during the smallpox outbreak of 1721. This data set has 6,224 observations, each representing an individual. Each observation collects data on two variables. The first first variable is Result which notes whether the individual lived or died in the outbreak. The second variable is Inoculated which notes whether or not that person was inoculated using a weakened strain of the smallpox virus.

Part I: Build a contingency table

Using the data matrix provided, use the COUNTIFS() function to complete the contingency table that is embedded in the spreadsheet.

Part II: Compute the joint probabilities

First we want to compute each disjoint probability within our data. This will allow us to get a sense of how likely each outcome is within our dataset. Then we can use the marginal and conditional probabilities to dive in a bit deeper. Complete the joint probability table embedded within our spreadsheet.

Part II: Compute the marginal probabilities

We will next use our contingency table to calculate the marginal (single variable) probabilities within our dataset. Fill out each probability in the marginal probabilities table embedded in the spreadsheet. How likley is it that someone who contracts smallpox dies?

Part IV: Compute the conditional probabilities

Next we want to compute the conditional probabilities found in our data. This will help us answer the question: did people who were inoculated have a higher chance of surviving smallpox than those who did not get inoculated? Complete the conditional probability table embedded within the spreadsheet.

Part V: Analyze the results

Finally we want to decide what the data is telling us. The most important question we can answer in this data is: Does inoculation substantially increase someone’s chance of surviving smallpox?

  1. How likely is it that someone dies of smallpox conditional on not being innoculated?
  2. How likely is it that someone dies of smallpox conditional on being inoculated?
  3. If there was a change in the percent of people who died of smallpox between the group who was inoculated and the group who was not inoculated, was it substantial? Explain, and be sure to define what you think substantal means in this context.