In the previous exercises, we looked at an association between the `influence`

and `leader`

questions using a contingency table. We saw some evidence of an association between these questions.

Now, let’s take a moment to think about what the tables would look like if there were no association between the variables. Our first instinct may be that there would be .25 (25%) of the data in each of the four cells of the table, but that is not the case. Let’s take another look at our contingency table.

```
leader no yes
influence
no 0.271695 0.116518
yes 0.212670 0.399117
```

We might notice that the bottom row, which corresponds to people who think they have a talent for influencing people, accounts for 0.213 + 0.399 = 0.612 (or 61.2%) of surveyed people — more than half! This means that we can expect higher proportions in the bottom row, regardless of whether the questions are associated.

The proportion of respondents in each category of a single question is called a *marginal proportion*. For example, the marginal proportion of the population that has a talent for influencing people is 0.612. We can calculate all the marginal proportions from the contingency table of proportions (saved as `influence_leader_prop`

) using row and column sums as follows:

leader_marginals = influence_leader_prop.sum(axis=0) print(leader_marginals) influence_marginals = influence_leader_prop.sum(axis=1) print(influence_marginals)

Output:

```
leader
no 0.484365
yes 0.515635
dtype: float64
influence
no 0.388213
yes 0.611787
dtype: float64
```

While respondents are approximately split on whether they see themselves as a leader, more people think they have a talent for influencing people than not.

### Instructions

**1.**

The solution code from the previous exercise has been provided in `script.py`

to create a contingency table of proportions (saved as `special_authority_prop`

) for the `special`

and `authority`

columns. Use this to calculate the marginal proportions for the `authority`

variable and save the result as `authority_marginals`

.

Print out `authority_marginals`

. Do more people like to have authority over people or not?

**2.**

Use `special_authority_prop`

to calculate the marginal proportions for the `special`

variable and save the result as `special_marginals`

.

Print out `special_marginals`

. Do more people see themselves as special or not special?