We covered many things: Using Z and t charts; Central Limit Theorem and Sampling Distributions; Confidence Intervals; Hypothesis Testing. PLEASE READ 8.1 and 8.2.
We cover two sample comparisons using large samples, small samples, pooled variances, unpooled variances.
Work these: 8.2, 8.3, 8.5, 8.8, 8.9, 8.11, 8.13, 8.15
We learn how to detect when an experiment should be seen as paired and when it should be seen as independent. We learn the impact of pairing.
Work these problems: 8.26, 8.29, 8.31, 8.34, 8.35
For 8.4 work: 8.43, 8.44, 8.46, 8.47, 8.54
For 8.6 work: 8.73, 8.74, 8.77, 8.78, 8.81
For 9.1 work: 9.1 through 9.8, 9.13
Variance between groups vs variance within groups.
Let's build out a chart on the board, work 9.17, 9.19 together.
Practice with: 9.20, 9.25, 9.18, 9.26, 9.33, 9.29
Overall error rates vs individual.
Practice: 9.35-9.39, 9.43, 9.45, 9.46
Practice: 9.64 - 9.70, 9.73
Work 9.105 completely and grade the write-up
Randomized block design is the many treatment version of paired difference.
Work these: 9.50-9.53, 9.59
Practice high level ANOVA ideas with these: 9.82-9.85, 9.91, 9.96, 9.102, 9.104
This is the many-bucket version of the binomial experiment. We call it goodness of fit because it is a test of how well a proposed distribution matches the distribution in a sample.
We'll talk out the details and practice with Skittles.
Problems to work: 10.1-10.5, 10.8, 10.9, 10.18
This is the two-factor multinomial experiment. The goal is to see if the proportions of one factor stay the same for all levels of some other factor. If not then there is an interaction. You hear these in headlines all the time, for instance: men that drink green tea are half as likely to get prostate cancer.
The sampling distribution is a Chi-squared, built just like the goodness of fit test, but with degrees of freedom (r-1)(c-1) and the expected values for each square being a perfectly independent prediction. That is, take the proportion for each row and column and multiple those by n.
I brought in a handout to work through.
Then we'll work these problems: 10.19-10.23, 10.25, 10.26, 10.33
This is the first time we get to take quantitative input and quantitative output. For a continuous range of independent/predictor values we want to make a best guess for an output. This is very useful, but it only applies inside the range of values we have observed. We'll get some statistical analysis cranking tomorrow.
Our best guess prediction will be a least-squares fit of our data plus an error term which will be an independent random variable centered around 0.
Practice line-math with these: 11.1-11.7
Practice least-squares with these: 11.10-11.12, 11.14-11.16
Problems to work: 11.27, 11.29, 11.32, 11.34
Problems to work: 11.41c, 11.45, 11.46 (careful), 11.48
Problems to work: 11.59, 11.63, 11.64, 11.67
11.79, 11.81, 11.82
We want to do overall fitness testing of a multiple regression model. Make predictions with the model using CI and PI. Test individual coefficients (rarely and a priori). Check the four conditions on the error distribution.
Test ideas with: 12.1, 2, 4, 9, 11; 12.25, 28; Explain figures 12.35, 12.36, SIA12.10 vs SIA12.13
Learn how to think about the twisted plane created by interaction terms. Typically analyze at fixed values for all but one input, then think about how the picture changes.
Practice with 12.34, 35, 37, 39, 38, 44
If we want to model data that has curvilinear patterns then we can add higher-order terms. At the moment we'll just use quadratic terms. An x^2 will be the curvature term, with positive coefficients indicating concave up and negative concave down. The linear terms will just be shifts. For multiple inputs we should add interaction terms too.
Practice with 12.49-53, 55, 56
We can model using qualitative variable (bucket variables) in a regression too. To pull this off we create many variables that will be valued either 0 or 1 depending on the actual input. So if the bucket variable is music genre and it takes three values: "Rock", "Reggae", "Elevator" we might create two dummy variables to describe this one attribute. They would be x_1 is 1 when "Reggae" and 0 otherwise and x_2 is 1 when "Elevator" and 0 otherwise. The model would look like E(y) = b_0 + b_1 x_1 + b_2 x_2. Notice that when Rock is the input value x_1 and x_2 will be 0 so E(y) = b_0. Etc.
Practice with problems 12.66-69, 72, 76
Study session in Lerner Trading Lab (first floor of Purnell with the big ticker) Wednesday 2-4ish.
If we have a qualitative variable with k levels and we want to mix with linear models then we will need k different lines. To achieve this we'll have a base-line line and then for each dummy variable an adjustment line. Same will be true for a quadratic model / dummy variable mix. Let's do a dummy variable for gender (which has two levels), x1 = 1 if male 0 otherwise, and a qualitative variable x2 = age. If we are trying to predict salary from age and gender then we would have a model like: E(y) = b0 + b1*x2 + x1*(b2 + b3*x2 ) which expands to b0 + b1*x2 + b2*x1 + b3*x1*x2.
Let's practice with: 12.82-12.86, 12.87, 12.89, 12.94
Now that we've learned to model data with many inputs, interactions, and second order terms we might want to test which of two tests does the better job. It will ALWAYS be that adding terms to a model will reduce error. So in our nested test we compare the amount of error that is reduced for each extra term against the error in the complete model. One of the key skills will be recognizing which subset of the coefficients you are testing for importance.
Let's practice with: 12.97, 98, 99, 101, 103
Study session in Lerner Trading Lab (first floor of Purnell with the big ticker) Today 2-4ish.