# On Confidence Intervals and Probabilities

# The problem

My professor made a statement today which I really hated. We were chatting about
confidence intervals. Suppose we have some normally distributed population
`p ~ N(10, 1)`

with a mean of 10 and a standard deviation of 1. Let `X`

be a
size-100 random sample from `p`

. Suppose we compute the 95% confidence interval
`I = (u, v)`

for the mean of `p`

. The question is: what is the probability that
`mean(p)`

is inside `I`

?

Let’s look at an example:

```
library(BSDA) # For Z test
set.seed(8)
p <- data.frame(val=rnorm(1000000, 10, 1))
mean(p$val)
sd(p$val)
N <- 100 # Sample size
X <- sample(p$val, N, replace=F)
z.test(X, sigma.x=1, mu=10, conf.level=0.90) # 90% CI, I1
z.test(X, sigma.x=1, mu=10, conf.level=0.95) # 95% CI, I2
```

This outputs the following (truncated) text describing two z-Tests, one with a 90% confidence interval and one with a 95% confidence interval.

```
[1] Population mean: 9.99916568117066
[1] Population std: 9.99916568117066
One-sample z-Test
data: X
z = 1.8306, p-value = 0.06716
alternative hypothesis: true mean is not equal to 10
90 percent confidence interval:
10.01857 10.34754
sample estimates:
mean of x
10.18306
One-sample z-Test
data: X
z = 1.8306, p-value = 0.06716
alternative hypothesis: true mean is not equal to 10
95 percent confidence interval:
9.987062 10.379055
sample estimates:
mean of x
10.18306
```

For simplicity, here is the population mean and the confidence intervals:

```
[1] Population mean: 9.99916568117066
90 percent confidence interval:
10.01857 10.34754
95 percent confidence interval:
9.987062 10.379055
```

So Prof asks “What’s the probability that the population mean is in the 90% confidence interval?” and everyone says “90%”. Prof is all “lulz nah it’s 0”, and we’re like “well yeah, obviously, but you know what we mean..” and that’s when he drops one on us: he says that a 90% confidence interval doesn’t have a 90% chance of containing the population mean. It has either a 0% chance or 100% chance.