Confidence intervals

The first fundamental tool for inference is the confidence interval. Here is the basic idea.

• We grab a random sample of \(n\) individuals from a population.
• We compute some statistic \(s\) from that sample -
  • That statistic might be a proportion \(\hat{p}\), a mean \(\bar{x}\), or almost anything.
  • Following the book, we’ll focus on proportions here.
• We compute a standard error \(SE\) for the statistic - more on that in a min.
• The confidence interval is then \[s \pm z^* \times SE.\]
  • The bigger \(z^*\) is, the bigger the interval is and the more confident we are that the interval contains the actual parameter.
  • Often \(z^*=2\) and we call the confidence interval a 95% confidence interval.

Standard error

The standard error is simply the standard deviation associated with a sampling distribution. Generally, if the standard deviation associated with a population parameter is \(\sigma\) and we form a random variable \(S\) by sampling from the population, the standard deviation of \(S\) is \(\sigma/\sqrt{n}\). Thus, as expected, the standard error decreases as the sample size increases.

Example

Suppose we draw a random sample of 132 people and find that 16 of them have blue eyes. Use this data to write down a 95% confidence interval for the proportion of people with blue eyes

Solution: We have \(\hat{p}=16/132 \approx 0.1212\) and \[SE(\hat{p}) = \sqrt{(16/132)\times(116/132)/132} \approx 0.02840718.\] Thus, our confidence interval is \[0.1212 \pm 2\times0.0284 = [0.0644, 0.178].\]

Margin of error

If you read the details of political surveys, you’re likely to come across the term “margin of error” at some point. Five Thirty Eight, for example, maintains a running Trump approval rating page. The page also points to poll details for a slew of polls. Check out the first one, namely the Gallup poll. There, we read “Daily results are based on telephone interviews with approximately 1,400 national adults; Margin of error is \(\pm 3\) percentage points”. What’s that mean?

Definition

When we write a confidence interval as \[s \pm z^* \times SE,\] Then, \(z^* \times SE\) is the margin of error. Geometrically, it’s the distance that the interval extends in either direction from the measured statistic \(s\).

Computation

So, where’s the \(\pm 3\) come from?

Suppose we’re writing down a confidence interval for a proportion. In this case, approve or disapprove. If the actual proportion is \(p\) and our sample size is \(n\), then the standard error is \[\sqrt{\frac{p(1-p)}{n}}.\] In our case, the take \(n\approx 1500\). Furthermore the biggest that \(p(1-p)\) can be is \(1/4\). You can see this by taking a look at a graph:

Thus, our standard error is at most \[SE \leq \sqrt{\frac{1/4}{1500}} \approx 0.01290994.\] Now, for a \(95\%\) confidence interval, we take \(z^* = 2\) so that our margin of error is at most \[ME \leq 2*0.01290994 \approx 0.026,\] which is rounded up to 3 percentage points.

This is a common thing to shoot for in political polls, which is why you often see sample sizes close to 1500.

A hypothetical poltical race

In a political contest, where there are two candiates and victory requires a simple majority, a candiate likes to be more than 3 percentage points above 50%. Can you see why?

An actual political race

In early November of last year, FiveThirtyEight reported that Donald Trump was only 3.3 perecentage points behind Hilary Clinton - almost within the margin of error. In fact, Clinton ended up winning the popular vote by 2.1%.

A poll on the first admendment

A recent poll by the Brookings Institute asks the following question of 1500 college students: “Is hate speech constitutionally protected?”

Here are the results:

		Political Affiliation			Type of College		Gender
	All	Dem	Rep	Ind	Public	Private	Female	Male
Yes	39	39	44	40	38	43	31	51
No	44	41	39	44	44	44	49	38
Don’t know	16	15	17	17	17	13	21	11

Use this to write down a confidence interval for the percentage of students who believe that hate speech is not constitutionally protected.

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