Stat 601, Fall 2000, Class 4

What you need to know from last time

Confidence intervals

1. A range of feasible values for an unknown population parameter, e.g. $\mu$ or $\sigma^2$

2. A statement conveying the confidence that the range of feasible values really does include the unknown population value

Sampling

What makes a good sample?

Obvious potential biases e.g. non-response

Precision as a guide to sample size

Making decisions

Hypothesis testing

Deciding on one of two choices

Null hypothesis: status quo

Alternative hypothesis: the converse of the null

Example; jury trial. Null is Innocent. Alternative is Guilty

Note - one is taken as true a priori

Decision based on collecting data - the jury votes. If jury votes = 12 then convict else acquit and declare NOT GUILTY. Note, do not declare innocent!

Two types of error

Innocent, but declare guilty (null true but go with alternative - Type I)

Guilty, but say innocent (alternative true but go with null - Type II)

Price of the errors? Which is worse (think capital trial)

What should error rates be?

Beyond all reasonable doubt - very small chance of incorrectly declaring guilty - small chance of a Type I error

The preponderance of the evidence

Criminal vs. civil court - context, cost dependent.

Hypothesis tests on means

All todays tests are standard error counters

How many standard errors is the null hypothesis mean away from the sample mean

If the null hypothesis mean is many standard errors (typically greater than 2) away from the sample mean, then the observed data is not in accordance with the null hypothesis, and we believe the data and reject the null

Types of test

One sample t-test; testing a single population mean, p.131

Two sample t-test; assuming equal variances, p.141.

Two sample t-test; NOT assuming equal variances, p.146.

Assumptions within groups

Independence

Constant variance

Approximately normal

The p-value

A measure of the credibility of the null hypothesis

Small p-values give evidence against the null

In English; the probability that if you did the experiment again and the null hypothesis was true, that you would observe a value of the bf test statistic as extreme as the one you saw the first time.

It picks up the repeatability idea. If something is true (ie the null hypothesis) then you should be able to replicate the observed results. A small p-value says that it would be hard to replicate, hence the small p-value offers evidence against the null

The paired t-test

The idea; two repeat observations on the same experimental unit

Twins, feet etc

Controls for unwanted variability between subjects

Essentially a one sample t-test on the differences