Recap Lecture 9 Basics Discrete distributions Binomial Poisson
Recap Lecture 9
Basics • Discrete distributions: • Binomial • Poisson • Continuous distributions: • Normal • Log-normal • Plots: • Histogram • Box plot • Q-Q plot
Sampling and estimation
Sampling of the Mean
Sampling of the Variance
Sampling of the proportion
How accurate are these estimates?
Confidence Interval for the Mean
A 95% approximate interval for a proportion Assume normality BUT WHAT IF THIS INTERVAL CONTAINS ZERO ? This would be possible if n is small, if p is nearly zero or if p is nearly one.
Logit-transformation Assume normality To get a 95% CI for p, we use the expit-transformation Now we are happy!
Basics of hypothesis testing • Null hypothesis • Alternative hypothesis – One-sided – Two-sided • Type I errors: Rejecting falsely • Type II errors: Accepting falsely
Level of significance So we want to construct a way to decide to • ACCEPT or • REJECT the hypothesis based on data in a way such that
Critical Region Assume • We want to test if the sodium content here is approx 3. 8 units • We have data y 1, …, yn • We have calculated average and SE. Support that content is < 3. 8 Support that content is > 3. 8
What do we know? If the content is 3. 8 then the average is normally distributed with mean 3. 8 With probability of 95% is the average less than 2*SE from 3. 8 If the true content is 3. 8 then the average is in the red area with prob 5%
P-value
Comparing means of groups • Two-sample problems – Paired (before-after setup) – Unpaired (independent setup) – Nonparametric: Mann-Whitney (independent setup) • K-sample problems – ANOVA (use post hoc analysis) – Nonparametric: Kruskal-Wallis • Equal variances?
Other non-parametric tests • K-S test • Chi-squared test • Binomial test
- Slides: 17