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70-208: Regression

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70-208: Regression

Lecture 1: Introduction to Regression Analysis

Spring 2014

John Gasper

- What is Regression? Why should we care? What can we do with it?
- How much do sales increase with every advertisement placed?
- How do wages of employees depend on education?
- How will the price of a stock change?
- Estimating demand (optimal pricing)
- Estimating effectsandPrediction/Forecasting

Teaching staff:

- Who am I?
- Who are you? Stop by my office.
- Office hours: Mon / Wed: 1-2pm and 4:30-5:30pm
- And by appointment

- Office hours: Mon / Wed: 1-2pm and 4:30-5:30pm
- Teaching Assistants for the course:
- TA: Adriana Lopez ([email protected] ) (CMUQ 1171)
- Office hours: by appointment only

- Undergrad Course Assistants: office hours TBA
- Syed TanveerHaider, AkhmedSungurov, FlavioFenley, Noor-Ul-Huda Admaney, and TanzeelHuda

- TA: Adriana Lopez ([email protected] ) (CMUQ 1171)

- Textbooks:
- Statistics for Business (main text; you should have it)
- Next Generation Excel (supplemental text – on reserve in Library)

- Attendance and participation
- Required. Clickers – bring them to every class.
- Blackboard + Piazza discussion site

- Cell phones and laptops
- Turn off your phones.
- Computers OK for for taking notes and working through data. NOT OK to check news, facebook, twitter, youtube…
- Seriously. If I or a TA sees you, odds are that I’ll ask you to leave.It’s disrespectful to me and other students.

- Grades: (aka what you stress over but shouldn’t)
- How do you get a good grade in this class?
- The only way to learn the material is to do it.
- Homework Exercises = 7%
- Problem Sets graded on Check System.

- Lab Quizzes (x5) = 4% each (20%)
- Attend 95% of classes and scored best 4 of 5.

- Midterm Exams (x3) = 15% each (45%)
- Final Exam = 25%
- Participation = 3%
- Attendance (clickers) + Discussion site (Piazza).

- Academic Integrity

- Warning: There is a lot of material in the course and we’ll move quickly.
- Any questions?

Data: what is it?

- Types of measurements: nominal, ordinal, interval, and ratio
- Categorical data
- Measures of Centrality: median, mode

- Numerical
- Measures of Centrality: median, mode, mean
- Measures of Spread: variance/standard dev, range, interquartile range, etc

- There are many ways to describe and examine data, and that at a basic level is what we’ll be doing in this class.
- You should be familiar with:
- Categorical
- 1 variable: bar charts, pie charts, etc.
- 2 variables: Contingency tables (x-tabs); Chi-sqtests

- Numerical
- 1 variable: histograms, boxplots, cumulative distribution
- 2 variables: scatterplots, correlation, t-test, etc…

- Categorical

- Histogram, PDF and CDF of exam scores:
- Scatterplot of Exam 1 and Exam 2:
(different class)

?

{

boxplot

{

histogram

Center: Median?

- 3.5

Inter quartile range?

- First to third quartile

Center: Mean?

- 3.8 Why?

Center: Mean?

- The mean is greater than the median here because the data are slightly skewed 3.8 vs 3.5

- There will be a review session this Wednesday / Thursday from 12:15-1:15
- Adriana will remind you how to use Excel to generate these graphical displays and quantitative summaries.
- 9am class (section W): Wednesday12:15 - 1:15
- 10:30am class (section X): Thursday 12:15 - 1:15
- Computer cluster 1185

- Come on time. If you’re late, you’ll be asked to leave.
- You can pick up your clickers at the review session
- While I can’t require you go, I would stronglyrecommend it. I won’t be slowing down to go over this stuff again
- Homework 1 (distributed today) is a review – you should have seen it before and I won’t cover it during class.
- Due 1 week from today

- Adriana will remind you how to use Excel to generate these graphical displays and quantitative summaries.

- What does ‘P(heads) = .5’ mean?
- What about ‘P(“Alice will get an A in Regression”) = .75’?
- Frequentistvs Bayesian interpretations. Differences don’t matter for this class and I’ll use language from both.

- Basic properties:
- 0 ≤ P(A) ≤ 1
- P(A) = 1 – P(Ac)
- P(A or B) = P(A) + P(B) – P(A and B)
- Events A and B are independentif the occurrence of one doesn’t tell you anything about the occurrence of the other.

- P(A and B) is often called the “joint probability”
- P(A) is the “marginal probability”
- P(A and B) + P(A and ~B) = P(A)

- The conditional probability
- P(A|B) = P(A and B) / P(B)
- P(A|B) is very different than P(B|A).

What is the Normal distribution?

- Often called the “Bell Shaped Curve.”
- This isn’t quite right. It is bell shaped, but there are many bell shaped distributions that aren’t the Normal dist.
Normal, or Gaussian, distributions are going to be very important for us.

- Often we’ll need to assume that a random variable X is Normally distributed, denoted X ~ N(μ,σ2)

Different μ

Different σ

- Random doesn’t mean haphazard. Consider an uncertain investment: X
- X could lose 1000 (with probability = .3)
- X could gain 10000 (with probability = .2)
- X could gain 100 (with probability = .5)

- X is a Random Variable. What is the expectation of X?
- E(X) = p(x1)x1 + p(x2)x2 + …p(xn)xn
- E(X) = 0.5*100 + 0.2*10000 + 0.3*-1000 = 1750 = μ

- Variance of X?
- Var(X) = E(X – μ)2 =σ2
- = (x1– μ)2 p(x1) + (x2– μ)2 p(x2) + … + (xn– μ)2 p(xn)
- = (100- 1750)2 * 0.5 + (10000 – 1750)2 * 0.2 + (-1000 – 1750)2*0.3

- And higher order moments Skew, Kurtosis, etc.
- Regression is basically about Conditional Expectation: E(Y|X)
- I.e., what do we expect about Y given we have some information X

Normality

- Why assume Normality? The Central Limit Theorem tells us that we’re often OK:
- The probability distribution of a mean (or sum) of IID random variables of tends to a Normal distribution (asymptotically)
- Several versions of the CLT but we won’t go through the proofs here (they can be a little nasty)
- So why are we OK?
- Observed data are often (not always) the accumulation of many small factors (e.g., the value of the stock market depends on many investors, or scores on an exam)

- A visual check on Normality
- Why wouldn’t just looking at the density or histogram work?
- Sometimes skew, kurtosis, etc, is easy to see but often it is not unless you look at a quantile plot

- Why wouldn’t just looking at the density or histogram work?

If data track the diagonal line,

you can safely assume it’s a

Normal distribution.

What is a z-score?

- Transforms a variable to standard deviation units away from the mean. Centered at 0.
- Why would we use it?

- What is P(X = 600)?
- What is P(X >= 600)?

- The lifetimes (in km) of a certain brand of automobile tires is a normally distributed random variable,
- X ~ N(μ=40,000 km, σ=2000 km)
- In a shipment of 3000 tires how many tires are expected to have a lifetime that is less than 35,000 miles?
- E(# of tires) = P(X < 35000) * 3000
- So how do we calculate P( X < 35000)?
- Z-scores. Or very easy in Excel: NORM.DIST()
- norm.dist(x, μ, σ, Cumulative?)
- norm.dist(35000, 40000, 2000, TRUE) = .0062
- E (# tires) = .0062 * 3000 = 18.6 = 19

- Z-scores. Or very easy in Excel: NORM.DIST()

- If any of the topics today seem hazy, review those chapters (take note of chapters 4, 12, and 15).
- Problem Set 1 due next Monday 9am.
- First quiz next Wednesday
- Pick up your clicker this week
- In the Excel review (9am Wednesday, 10:30 Thur)
- Sunday 10:30-11 in Adriana’s office.
- Must have it by next Monday’s class

- Excel reviewon Wednesday and Thursday (depending on section)
- Don’t come late – it’s distracting – we’re starting at 12:15 to give you a 25min break for lunch.
- You’ll be asked to leave if you’re late. Again, it’s very distracting for the students who came on time.