Confidence intervals

We motivated CI’s as all values of \(\mu_0\) that would not be rejected in a two-sided hypothesis test of \(H_0: \mu = \mu_0\).

Two-sided p-values, two-sided rejection regions and two-sided confidence intervals are generally equivalent: \[ \begin{aligned} p < \alpha &\iff \text{Reject } H_0: \mu = \mu_0 \text{ at level } \alpha \\ &\iff \mu_0 \text{ is outside} (1-\alpha)100 \text{\% confidence interval} \\ p > \alpha &\iff \text{Fail to reject } H_0: \mu = \mu_0 \text{ at level } \alpha \\ &\iff \mu_0 \text{ is inside} (1-\alpha)100 \text{\% confidence interval} \end{aligned} \]

One-sided confidence intervals

You can motivate them from one-sided tests too.

You end up with an infinite bound on one end.

So, why not report one sided CIs?

You don’t always do a hypothesis test. A plausible range for a parameter value should be two-sided. (If there isn’t a value of interest, how could there be a direction of interest?)

Should a plausible range for depend on your hypothesis of interest? More useful for others to give a 95% two-sided interval.

Yes, this means your one-sided test might not agree with your two-sided confidence interval.

Should we ever do one-sided tests? Some people argue “No, we should never do one sided tests”. I’d say, you can, but you better have a really good reason, or someone will accuse you of doing it just to get a smaller p-value.

Data Setting

Population: \(Y \sim \text{Bernoulli}(p)\), i.e. \[ Y = \begin{cases} 1, & \text{with probability } p \\ 0, & \text{with probability } 1 - p \end{cases} \]

\(E(Y) = p\), \(Var(Y) = p(1-p)\) When mean and variance share parameters we say there is a mean-variance relationship.

Parameter: \(\mu = E(Y) = p\), the population proportion

Sample: \(n\) i.i.d from population: \(Y_1, \ldots, Y_n\)

Statistic: \(\overline{Y} = \frac{1}{n}\sum_{i=1}^n Y_i = \hat{p}\), the sample proportion.

Testing

Null hypothesis: \(H_0:p = p_0\)

1. Exact test: use fact that \[ n\overline{Y} \sim \text{Binomial}(n, p) \]

2. Approximate test: use fact that \[ \overline{Y} \dot\sim N\left( E(Y) , \frac{Var(Y)}{n}\right) = N\left( p , \frac{p(1-p)}{n}\right) \]

Exact Binomial Test:

Complete worksheet (Charlotte will provide)

1. Get into groups according to number on worksheet at numbered whiteboard

2. Write answers to bold questions on whiteboard as you complete them (so I can see where you are up to)