STATS 250

Confidence Intervals for One Population Mean

Standard deviation of the sample mean

$$\text{s.d.}(\bar{x}) = \frac{\sigma}{\sqrt{n}}$$

This is approximately the average distance of the possible \(\bar{x}\) values (for repeated samples of size \(n\)) from the true population proportion \(\mu\).

The issue here is that you typically do not know \(\sigma\) in advance.

Instead, use the sample standard deviation \(s\) to estimate the population standard deviation.

Standard Error of a Sample Mean

$$\text{s.e.}(\bar{x}) = \frac{s}{\sqrt{n}}$$

This estimates the difference approximately.

Z-Statistic for a Sample Mean

$$z = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}}$$

However, the dilemma is that you rarely know sigma. So you replace the standard deviation with the standard error.

$$z = \frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}}}$$

This notably does not have a normal distribution. Instead, it has a t-distribution with degrees of freedom of \(n-1\).

T-Distributions

  • Symmetric, unimodal, centered at 0.
  • Flatter with heavier tails compared to \(N(0,1)\) – there are higher probability of outliers.
  • As \(df\) increases, The distribution begins to approximate the normal distribution.

Every statistic has a sampling distribution, but the distribution may not always be normal or bell-shaped.

CIs for means

An interval is a range of reasonable values for the parameter with a high level of confidence.

The Central Limit Theorem states that, as the sample size gets large, the sampling distribution approaches:

$$N(\mu, \frac{\sigma}{\sqrt{n}})$$

This makes the confidence interval:

$$\bar{x} \pm t^* \text{s.e.}(\bar{x})$$

The rule of thumb is that, if you're finding a CI for a proportion, use a \(z\). For a mean, use a \(t\). If the sample size is large (n > 30), then it begins to approximate the normal distribution. Make sure to always round down the degrees of freedom, if there isn't a match. That gives you a large bound of error.

Important things to do

  • Be sure to check the conditions
    • Is it normal?
  • Interpret the CI in the context of the problem
  • Be able to explain the confidence level