8: Confidence Intervals and Significance Tests

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.title[
# 8: Confidence Intervals and Significance Tests
]
.subtitle[
## Empirical Methods
]
.author[
### <large>J. Seawright</large>
]
.institute[
### <small>Northwestern Political Science</small>
]
.date[
### Oct. 9, 2025
]

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### Attitudes about Violence

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Looking just at the most recent results, how confident should we be that the 67%
for Republicans is really higher than the 58% for Democrats? Or is it possible this
is a coincidence of sampling?

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* We can use statistics to ask: if we generated samples at random, how often would 
we get samples of this size where the difference between the two groups was at least
this big if the true underlying difference was zero?

* There are about 1000 respondents in each party at each time period.

---

``` r
prop.test(c(580,670),c(1000,1000))
```

```
## 
## 	2-sample test for equality of proportions with continuity correction
## 
## data:  c(580, 670) out of c(1000, 1000)
## X-squared = 16.898, df = 1, p-value = 3.944e-05
## alternative hypothesis: two.sided
## 95 percent confidence interval:
##  -0.13325075 -0.04674925
## sample estimates:
## prop 1 prop 2 
##   0.58   0.67
```

---

For another comparison, how confident should we be that the 67% of Republicans who
see political violence as a serious problem in September of 2025 is higher than the 44% with
the same opinion in June of 2025?

---

``` r
prop.test(c(440,670),c(1000,1000))
```

```
## 
## 	2-sample test for equality of proportions with continuity correction
## 
## data:  c(440, 670) out of c(1000, 1000)
## X-squared = 106.17, df = 1, p-value < 2.2e-16
## alternative hypothesis: two.sided
## 95 percent confidence interval:
##  -0.2733778 -0.1866222
## sample estimates:
## prop 1 prop 2 
##   0.44   0.67
```

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An **observational study** is a research project in which we don't get
to assign values on the independent variable. Instead, we just observe
what happens in the normal course of events.

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### Observational Studies

-   Observational studies can sometimes be quite good at external
    validity.

-   However, confounding variables are a major problem for internal
    validity.

-   Such studies might be strongest at answering questions other than
    "why" questions.

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### Large-N Methods

**Large-N methods** are the statistical approach to analyzing
observational studies.

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### Large-N Approaches to Political Violence

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