2.1 T-test

1. You have 2 populations.

2. Do they have the same mean?

3. Take a sample of observations from each population.

4. Calculate the sample means.

5. They're probably different.

6. What's the prob the sample means would be at least that different if the population means were the same.

7. At least can be 1 sided or 2 sided.

2.2 ANOVA

1. Analysis of variance

2. Test for possible difference in several groups.

3. E.g. you're searching for a cure for lycanthropy.

4. 5 possible treatments: aspirin, silver crosses, sunlight, being bitten by Dracula, nothing.

5. Take 100 people with lycanthropy.

6. Assign different treatments randomly.

7. Measure length of hair at next full moon.

8. Did any treatment work?

9. Real work application: The worldwide pharma industry grosses $$10^{12}$$ dollars a year. A new drug costs several $$10^9$$ to develop, including the costs of the failures. To get a new drug approved, you have to prove, with trials and statistics, that it works.

2.3 Linear regression

1. To explore possible linear relationships between several variables.

2. Several possible independent variables.

3. One dependent variable.

4. One independent variable example:

   1. student score vs time on exam 2:

   2. Independent variable: time to finish.

   3. Dependent variable: score.

   4. Is there a linear relationship?

   5. What is it?

   6. How good is it?

5. Multiple independent variables example:

   1. Try to predict first year student performance at RPI.

   2. Dependent variable: first year GPA.

   3. Independent variables:

      1. high school grade

      2. high school rank

      3. number of AP

      4. fraternity?

      5. athlete?

      6. home state

      7. height

      8. weight

   4. which one is the strongest predictor?

   5. Add the independent variables one by one in order of importance.

   6. However, independent variables may be correlated with each other.

   7. with enough independent variables you can explain anything.

   8. what about nonlinear relationships?

2.4 Non parametric stats

1. no assumptions about the distribution, except that the observations are independent.

2. Often use order stats.

3. E.g., Wilcoxon rank-sum test (aka Mann-Whitney) to test if two pops have same mean:

   1. combine the observations from the two populations, X and Y.

   2. sort them all together

   3. see if the observations from population X are clustered at the start.

   4. by computing U score: count number of times Xi>Yj,

      1. for large enough n, U is normal, with mean $$n^2/2$$ and variance $$n^2(2n+1)/12$$.

   5. what's the probability that observations would be this biassed (towards the start) if the population means were the same? I.e., that U would be this far off mean?

4. There are many tests.

5. You need to decide what "biassed" means. I.e., pick your alternative hypothesis.

6. Not as powerful but more robust.

2.5 How to lie with statistics

https://en.wikipedia.org/wiki/How_to_Lie_with_Statistics

https://www.amazon.com/How-Lie-Statistics-Darrell-Huff/dp/0393310728

2.6 Machine learning

current hot application of stats.