What two options of samples must we consider?

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- Independent
- Paired

What is meant by independent samples?

Observation of X is unrelated to observation of Y, no association between samples

What is meant by paired samples?

Each X is "paired with" a similar Y, before\after measurements

What two distinct approaches can our method take?

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- Parametric
- Non parametric

What is meant by a parametric method?

Assume distributions for X, Y. Test difference in population means by comparing parameters.

What is meant by a non-parametric method?

No assumed distribution, no parameters to compare

What is the null hypothesis for an independent sample t-test?

No difference in means -

How do we estiamte μ_{x} - μ_{y} by in the independent sample t-test?

What is the variance of estimate for independent sample t-tests?

How is the following distributed?

How does the following simplify under the hull hypothesis H_{0}: μ_{x} - μ_{y} = 0?

When σ^{2} is unkown in the following what do we estimate it with?

Pooled sample variance

Define the **pooled sample variance.**

What are two properties of the pooled sample variance?

Prove the following property of the pooled sample variance.

Prove the following property of the pooled sample variance.

What is the standard error of the following?

What is the theorem about the independent sample t statistic?

Prove the following theorem.

What is the significance test of the hypothesis and confidence interval for the independent sample t-test?

What are three assumptions to do the independent sample t-test?

How do we check the following three assumptions?

What is independent in the paired sample t-test?

X, Y are correlated but each pair is independent.

What is the Cov[X,Y] in the paired sample t-test?

Cov[X,Y] = ρσ_{X}σ_{Y}

What does ρ stand for?

The correlation between X and Y

What do we look at in a paired smaple t-test?

The differences between each pair

What does D_{i }equal?

D_{i} = X_{i} - Y_{i}

What is the 𝔼[D_{i}]?

What is the Var[D_{i}]?

What distribution do the D_{i}'s have?

What do we estiamte μ_{X} - μ_{Y} by in the paired sample t-test?

Show that the following is an unbiased estimate.

What is the variance of the estimate in paired sample t-test?

What is the distribution of the estimate in the paired sample t-test?

What is the effect of correlation in the paired sample t-test?

Which means Var[Ď] < Var[Ẍ - Ȳ] (should all be bars), pairing the data and working with difference improves out precision.

What is theorem about the paired-sample t test statistic?

What is the significance test of the hypothesis and confidence interval for the paired sample t-test?

When the assumptions hold are nonparametric methods or parametric methods more powerful?

Parametric

What do we focud on in non parametric tests?

Ranking the data

What is a general null hypothesis for non parametric independent samples?

No group difference.

Define the **rank sum test.**

What do you do when you have tied observations in the rank-sum test?

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- Rank-sum test requires that ranking is unique
- So you assign the average of the tied ranks

What two options do we have for the distribtuin for W?

What value do the tables of W give?

TL

What is T_{L} and T_{U}?

The upper and lower critical values

How do you find T_{U} once you have found a value for T_{L} from the W tables?

When we have found T_{L} and T_{U} when do we reject H_{0}?

What is the theorem about the Normal approximation to W?

Prove the following theorem.

What is the method for the signed-rank test for paried samples?

Define **the signed-rank test.**

What is another name for the signed-rank test?

The Wilcoxon signed-rank test

How should d_{i} = 0 be dealt with in the signed-rank test?

Discard these points and reduce n accoridngly

What value of V^{+} or V^{-} can you get from the table of V values?

The smallest value

When do we reject H_{0} in the signed-rank test for paired samples?

What is the theorem about the Normal approximation to V?

Prove the following theorem.