Mathematical Statistics Lecture

A point estimate ( \hat\theta = 3.2 ) is useless without error bounds. A gives a range that covers ( \theta ) with a prescribed probability ( 1-\alpha ).

Assume ( X_i \sim \mathcalN(\mu, \sigma^2) ). We know that: [ Z = \frac\barX - \mu\sigma / \sqrtn \sim \mathcalN(0, 1) ] mathematical statistics lecture

If we want to know the average height of a population, we need a single "best guess" based on our sample. This is called a . A point estimate ( \hat\theta = 3

The probability that we would see our results (or more extreme results) if the Null Hypothesis were true. A low p-value (usually < 0.05) suggests we should reject the Null. 6. Confidence Intervals: Quantifying Certainty We know that: [ Z = \frac\barX -

While "Statistics" often refers to the collection of data, is the branch of mathematics that uses probability theory to analyze and interpret that data. 1. The Core Objective: Population vs. Sample

It is incorrect to say "There is a 95% probability that ( \mu ) lies in this interval." The parameter ( \mu ) is fixed; the interval is random. Correct: "95% of such intervals constructed from repeated sampling will contain ( \mu )."