Taking a sample in order to infer about the population:

• We want to know the population mean \(\mu\), so we take a sample mean \(\overline{x}\).
• But instead of using a single value \(\overline{x}\), why not a range of plausible values?

Imagine the \(\mu\) is a fish:

Sample Mean \(\overline{x}\)	Confidence Interval

Let's focus on the case of taking many, many, many samples of size n=5 and n=50 from the population of OkCupid users:

n <- 5
samples_5 <- do(10000) * 
  mean(resample(profiles$height, size=n, replace=TRUE))
samples_5 <- samples_5 %>% 
  as_data_frame()

n <- 50
samples_50 <- do(10000) * 
  mean(resample(profiles$height, size=n, replace=TRUE))
samples_50 <- samples_50 %>% 
  as_data_frame()

We computed

Let's construct 95% confidence intervals via two methods:

1. Looking at the middle 95% of data using quantile()
2. If the data is bell-shaped, using the rule of thumb

Recall from Lec28 Chalk Talk: quantiles. Run the following in your console:

x <- c(0:12)
x
quantile(x, prob=c(0.25, 0.5, 0.75))

This is saying

• ~25% of x values are less than 3
• ~50% of x values are less than 6
• ~75% of x values are less than 9

Behold: the normal distribution i.e. the bell curve.

Properties of the Normal Distribution:

• Unimodel, symmetric
• Entirely defined by its
  • mean \(\mu\): i.e. center
  • standard deviation \(\sigma\): i.e. spread
• Key: 95% of values lie within \(2\sigma\) of \(\mu\)
  • i.e. the interval \([\mu - 2\sigma, \mu + 2\sigma]\) contains ~95% of values

Below we have a Normal Distribution with \(\mu=5\) and \(\sigma=2\)…

… Interval \([\mu -2\sigma, \mu +2\sigma] = [5 -2\times 2, 5 +2\times 2] = [1, 9]\) contains 95% of values (in purple).

Construct 95% confidence intervals for \(\mu\), the true average height of all 60K OkCupid users

• Using both methods for constructing confidence intervals: quantiles and bell-curve aproach
• For both when n=5 and n=50

Hint: Look at the histograms of the 10000 simulations.