Stat 601, Fall 2001, Class 4

What you need to know from last time

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Confidence intervals - why make them, how to interpret them
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Sampling - what makes a good sample, when samples go bad
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Decision making - dichotomous decision setup

Hypothesis tests on means

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All todays tests are standard error counters
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How many standard errors is the null hypothesis mean away from the sample mean
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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
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Types of test
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One sample t-test; testing a single population mean, p.131
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Two sample t-test; assuming equal variances, p.141.
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Two sample t-test; NOT assuming equal variances, p.146.
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Two sample non-parametric tests; NOT assuming approximate normality, p.152. Median + Van der Waerden
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Assumptions within groups
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Independence
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Constant variance
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Approximately normal

The p-value

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A measure of the credibility of the null hypothesis
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Small p-values give evidence against the null
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In English; the probability that if you did the experiment again and the null hypothesis were true, that you would observe a value of the test statistic as extreme as the one you saw the first time.
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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

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The idea; two repeat observations on the same experimental unit
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Twins, feet, sales territories, similar firms etc
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Controls for unwanted variability between subjects
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Controlling for variability is a very powerful idea. It's like denoising the data, so that the signal is more apparent
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Essentially a one sample t-test on the differences

Putting ideas together

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Dealing with confounding variables
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Review example + introducing correlation

Dealing with confounding variables

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Marginal association - the relationship between two variables ignoring other possible explanatory variables
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Partial association - the relationship between two variables having taken into account other explanatory variables
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Association does NOT imply causation

Example Salary.jmp

Review example - FinMark.jmp

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Points to note
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Log transforms to linearize exponential growth
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Returns remove most time trend, reveal volatility
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Relative volatility between T-bills and VW-return
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Good normality properties but a bit fat tailed
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Return/variance tradeoff in portfolios

Exam discussion

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Format: similar to homework questions, but relevant output provided
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Expect to take not more than 2 hours
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Covered: chapters 1 - 8 in the Case Book
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Key concepts
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Descriptive summaries, mean, median, sd, variance
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The empirical rule
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The standard error of the mean
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Quality control charts
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Confidence intervals
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Sampling and assumptions
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Hypothesis tests on means, test statistics and p-values


Subsections

2001-10-05