1. Statistical Inference
Political Analysis 2, Week 5, Lab 4

2. What is inference? Sample: Population:
a subset of cases drawn from the pop.
Population:
all relevant cases of interest

3. What is inference? Sample: Population:
a subset of cases drawn from the pop.
Population:
all relevant cases of interest
Inference is the process of using what we know to be true about a sample to make statements about what is likely to be true in the broader population.

4. What is inference? Sample: Population:
a subset of cases drawn from the pop.
Population:
all relevant cases of interest
Inference is the process of using what we know to be true about a sample to make statements about what is likely to be true in the broader population.
Ex: ABC/Washington Post  Clinton (47%) to Trump (43%) to Johnson (4%); Nov. 3-6; 2220 Likely voters (sample)

5. Tools of inference
What are…
standard errors?
t-values?
p-values?

6. Inference involves uncertainty
How reliable are our findings?
Uncertainty shapes what conclusions we can infer about the population based on our sample
Two ways to assess uncertainty:
Simulations
Central limit theorem

7. Uncertainty, uncertainty, uncertainty…

8. Simulating a thought experiment
Suppose we take a random sample among a population of 50,000 individuals on a variable of interest (x1)
We want to know whether the sample mean is close to the true population mean
The magnitude of random error depends on:
a) sample size; and
b) true population mean

9. Simulating a thought experiment (2)
And suppose we know the true population mean of x1 = 20.
Using R, I can randomly draw a sample of 10 individuals from our population of 50,000

10. Simulating a thought experiment (3)
And we can draw another sample of 10 …

11. Simulating a thought experiment (4)
We can increase the sample size to 100 observations.

12. Simulating a thought experiment (5)
We can store our sample of 100 and take the mean.

13. Simulating a thought experiment (5)
We can store our sample of 100 and take the mean.
We can perform this function again.

14. We can do this 10,000 times (using a loop in R) and plot these means in a histogram.
 Sampling distribution: hypothetical distribution of sample means
 We can create a confidence interval for our histogram:
We know that 95% of the samples had a mean between 17.1 and 23:

15. Central limit theorem
CLT: If we take an infinite number of random samples,
and plot the sample means to each random sample,
those sample means would be distributed
normally around the true population mean.
i.e., the mean of the sampling distribution would
equal the true population mean
The estimate of the standard deviation of the sampling distribution is what we refer to as the standard error of the mean

16. Example: rolling dice Suppose we roll a dice 600 times
We learn that the mean value is 3.47
3. From our mean and sample size we can calculate the standard deviation, which is 1.71

17. Example: rolling dice (2)
To get the standard error of the mean…
standard error of the mean

18. Normal distribution 1 sd of mean: 68% 2 sd of mean: 95%
1. But first must standardize random variable!
2. And then look up probability
of finding value as distant
from mean as ours 
p-value!

19. t-distribution
For OLS regression, we use the t-distribution (because we don’t know the standard deviation).
Same logic as normal distribution:
1. Standardize the coefficient:
= t-value
2. And then look up
Probability of finding value
as distant as ours 
p-value!

20. Hypothesis testing The logic of hypothesis testing is:
Define a “null hypothesis” and an “alternative hypothesis”
Calculate your statistic (e.g., mean, correlation, regression coefficient)
Calculate the t-value
Find the associated p-value: the probability of getting a statistic as large as yours if the null hypothesis were true
Choose a threshold alpha: 0.1, 0.05, 0.01
If the p-value is lower than the threshold, we reject the null hypothesis and say that the coefficient is statistically significant.