Point Estimates

• \(\bar{x}\), \(s^2\), \(\hat{\beta_0}\)

  • We’ve only calculated these so far

• They’re all wrong estimates

  • How do we use a sample mean to infer the population mean?

  • Why did we spend 5 weeks on distributions and probability?

How old is the universe?

How old is the universe?

How old is the universe?

Frequentist

Methods using the perspective that the probability of an event is defined as the relative frequency of that event’s occurence across infinite trials.

• Empirical probability

• “Data driven”

  • Data dependent

• Central Limit Theorem

• Simple in application

Bayesian

Methods using the perspective that the probability of an event is the result of comparing the likelihood and prior knowledge of the event with the observed evidence.

• Prior knowledge

• “Assumption driven”

  • Open to assumptions

• Horrendous in application

Linear Regression

• Frequentist:

\[ \begin{aligned} y = \beta_0 + \beta_1x +\epsilon \\ \epsilon \sim N(0,\sigma^2) \end{aligned} \]

• Since \(\mathbb{E}(\epsilon) = 0\):

\[ y \sim N(\beta_0+\beta_1x,\sigma^2) \]

Linear Regression

• Bayesian model:

\[ \begin{aligned} y \sim N(\beta_0+\beta_1x,\sigma^2) \\ \beta_0 \sim N(0,\sigma^2_{\beta_0}) \\ \beta_1 \sim N(0,\sigma^2_{\beta_1}) \\ \sigma^2 \sim \text{inv-Gamma}(q,r) \end{aligned} \]

Bayes’ Rule

\[\text{Posterior}\ = \ \frac{\text{Likelihood} \times \text{Prior}}{\text{Evidence}}\]

\[P(A|B)=\frac{P(B|A)P(A)}{P(B|A)P(A)+P(B|A^c)P(A^c)}\]

Interpreting Intervals

• Confidence Intervals

  • Frequentist

  • “the probability the method is going to produce the correct result.”

• Credible Intervals

  • Bayesian

  • “the probability that the estimate is the truth.”

Sampling Distributions

Sampling Distributions

Intervals

Given \(\bar{x} = 53.17\), \(s=17.127\), and \(n=10\), calculate the \(95\%\) interval of deer body weights.

\(95\%\) z-interval: \[P(-z_0 < Z < z_0)=\ 0.95\]

Intervals

\[ \bar{x} \sim N(\mu,\frac{\sigma^2}{n}) \]

• We need population parameters to proceed

  • How can we get rid of them? Fix the distribution to \(N(0,1)\)?

• What could we do to fix a non-standard normal distribution?

Intervals

\[ z = \frac{x - \mu}{\sigma} \]

• This still needs population parameters

Pivots

Pivotal Quantity (Pivot): Any function who’s probability distribution does not rely of the population parameters of a given random variable.

\[t^*= \frac{x-\bar x}{s / \sqrt n}\]

• We can “free” the statistic from the population

• Then inference can come exclusively from samples

t-Distribution

t-Pivot

\[T=\frac{\bar x - \mu}{s / \sqrt n}\]

\[P(-t_0 < T < t_0)=\ 0.95\]

\[ \begin{aligned} P(-t_0 < T < t_0) =\ 0.95 \\ P(-t_0 < \frac{\bar x-\mu}{s / \sqrt n} < t_0) =\ 0.95 \\ P(-t_0 \ \frac{s}{\sqrt n} < \bar x-\mu < t_0 \ \frac{s}{\sqrt n}) =\ 0.95 \\ P(\bar x-t_0 \ \frac{s}{\sqrt n} < \mu < \bar x+t_0 \ \frac{s}{\sqrt n}) =\ 0.95 \\ \end{aligned} \]

Confidence Intervals

\[ \begin{aligned} 0.99= & \ 1-\alpha \\ \\ \alpha= & \ 0.01 \\ \\ \alpha/2 = & 0.005 \end{aligned} \]

\[\bar x \pm t_{\alpha/2} \ \frac{s}{\sqrt n}\]

Confidence Intervals

\[ \begin{aligned} \text{Point estimate} & = \bar x \\ \\ \text{Margin of error} & = t_{\alpha/2} \ \frac{s}{\sqrt n} \\ \end{aligned} \]

\[\text{Point estimate} \pm \text{Margin of error}\]

t-table

Sample Mean Intervals

\[ \begin{aligned} \text{CI}\ =& \ \bar x \pm t_{\alpha/2} \ \frac{s}{\sqrt n}\\ \\ \text{CI}\ =& \ 53.17 \pm 2.262 \times \frac{17.127}{\sqrt{10}}\\ \\ \text{CI}\ =& \ 53.17 \pm 12.25107 \\ \\ \text{CI}\ =& \ \begin{cases} 53.17 - 12.25107 = 40.91893\\ 53.17 + 12.25107 = 65.42107 \end{cases}\\ \\ \text{CI} \ =& \ (40.92,65.42) \end{aligned} \]