ORF 245: Statistics

ORF 245: Statistics

Fall '15

(Click here for Fall '14)

Lectures Based on Deveaux's Forthcoming Book

(Some) Lecture Slides:

Deveaux's Textbook Chapters:

Reading Assignments:

Wed, Sept 16, 2015: Chapter 1
Mon, Sept 21, 2015: Chapter 2
Mon, Sept 28, 2015: Chapter 3
Mon, Oct 5, 2015: Chapter 4
Wed, Oct 14, 2015: Chapter 5
Wed, Oct 28, 2015: Chapter 6
Fri, Nov 20, 2015: Chapter 7
Mon, Dec 14, 2015: Chapter 8

Homework Assignments:

Fri, Sept 25, 2015:
Chapter 1, Exercises 1-2, 3(d), 4-13
Additional Exercise: Using the data file BodyFat.txt, make a plot of Percent Body Fat vs. Age.
Fri, Oct 2, 2015:
Chapter 2, Exercises 3-7, 11, 12, 15, 16, 17
Fri, Oct 9, 2015:
   Chapter 3, Exercises 1, 6, 7, 9
   Additional Exercises:
1. Using the data from Figure 1.1 (available from the examples link below), compute the regression line \(\hat{y} = b_0 + b_1 x\). Plot it. Compare to the line produced by the call to lsline.
2. Suppose we have a list of numbers which we’ll call \(y\)-values for which there is no associated list of \(x\)-values. In this case, we can introduce the following (simple) regression model: \( y_i = \beta_0 + \varepsilon_i\). Find a formula for the least squares estimator \( b_0 \) for \( \beta_0\).
Fri, Oct 16, 2015:
   Chapter 3, Exercises 10, 12, 15, 16
   Chapter 4, Exercises 1, 4
   Additional Exercises:
1. Figure 2.11 shows a histogram of the distribution of systolic blood pressures based on a sample of 100 people.
  Figure 2.13 shows a histogram of the distribution of the sample means based on bootstrapping off the original sample 1000 times.
  For this exercise, use the same data set but instead of sampling 100 people from the population, sample just 30 people. And then do a bootstrap resampling from these 30 people 1000 times to make a new sample-means histogram. Compare this histogram to the one shown in Figure 2.13.
2. An Addition to Chapter 3, Exercise 10: using \( x = Sugar \) and \( y = Calories \):
  (a) Compute the correlation coefficient \( r \) between \( x \) and \( y \).
  (b) Compute the standard deviation of the regression residuals \( s \) using the formula on page 3-39.
  (c) Compute the coefficient of determination \(R^2\) using the formula on page 3-42.
Fri, Oct 30, 2015:
Chapter 5: Exercises 4, 5, 6(a,b,c), 7, 9, 12
Fri, Nov 13, 2015:
Chapter 5: Exercises 13, 16, 18, 23, 24, 27
Chapter 6: Exercises 2, 4, 6, 7, 8
Fri, Nov 20, 2015:
   Chapter 6: Exercises 9, 10, 11, 14, 16, 18, 23
   Additional Exercises:
1. Consider a coin that is biased. Suppose that we will toss the coin \( n = 50 \) times in an effort to determine an estimator \( \hat{p} \) for the true underlying probability \( p \) that the coin comes up heads. For \( i=1,2,\ldots,n, \) let \( X_i \) be a Bernoulli random variable taking the value 1 if the coin comes up heads and 0 otherwise. Here is the outcome from an actual experiment:
  \( x = [1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1] \)
  1. Compute the average value of these \( n \) Bernoulli values.
  2. Compute the standard deviation of these \( n \) Bernoulli values.
  3. Consider the Bayesian perspective. Suppose that the prior distribution for the unknown true probability \( p \) is uniform on \( [0,1].\) What is the formula for the posterior distribution? Plot this posterior distribution.
  4. Consider a bootstrap approach. Resample the 50 Bernoulli trials thousands of times (10,000 resamplings on my laptop computer goes fast) keeping track of the average value after each resampling. Plot a histogram of the average values.
  5. Make a plot of a Normal density in which the mean \( \mu \) is equal to the average value of the Bernoulli's (calculated in part (a)) and the standard deviation \( \sigma \) is equal to the standard deviation computed in part (b) divided by \( \sqrt{n}. \)
  6. Compare the plots in parts (c), (d), and (e). Comment on any similarities and notable differences.
2. Repeat all steps of the previous problem using this set of data:
  \( x = [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0] \)
Fri, Dec 18, 2015:
Chapter 7: Exercises 2(b), 4, 5(a), 6, 7(a), 8, 9, 10, 11(a,b,d,e), 14, 17, 19
Fri, Jan 8, 2016:
Chapter 8: Exercises 1, 2, 3, 4, 5, 6, 10

Data Sets:

Click here.

Examples Worked Out in Matlab:

Click here.

Online Forum to Discuss Class Topics: Piazza

Syllabus:

Statistics and Data: types, models, samples, populations, etc.
Displaying and Summarizing a Single Variable: categorical vs. quantitative
Samples, Resampling and Bootstrap
Displaying and Summarizing the Relationship between Two Variables
- Time Series
- Regression Models
Statistical Modeling: transformations, multiple regression, etc.
Probability: sample spaces, events, moments
Bayes' Rule
Continuous Random Variables
Important Random Variables: Bernoulli, Geometric, Binomial, Poisson, Normal, Exponential
Sampling Distribution of a Proportion
Sampling Distribution of a Mean – Central Limit Theorem
Estimation – unbiasedness and variance
Confidence Intervals
Hypothesis Testing: P-values, Alpha Levels, Significance, Power
Inference for Regression
Linear Regression
Time Series Components
Smoothing

Throughout the course, we will use the computer language Matlab to perform computations on real-world data to illustrate the methods and ideas covered in the course. The data sets will span from historical stock market prices to temperature data needed to study climate change.

Course Info:

Homeworks due every Friday, 5:00pm. 10% off for each day (or part of a day) thereafter.
At the end of the semester, the lowest homework grade will be dropped
Two midterms (Friday October 23 and Friday December 4).
Final grade based on:
Homework:
45%
Midterm 1:
15%
Midterm 2:
15%
Final:
20%
Participation:
5%
AIs: Koushiki Bose, Carson Eisenach, Huanran Lu, Elahesdat Naghib, Yuyan Wang.

Precepts: 3:30-4:20pm and 7:30-8:20pm on Mondays and Tuesdays.

Schedule:		Afternoons	Evenings
	Weeks 1 and 2:	Carson	Koushiki
	Weeks 3 and 4:	Carson	Elahe
	Weeks 5 and 6:	Huanran	Koushiki
	Weeks 7 and 8:	Yuyan	Huanran
	Weeks 9 and 10:	Yuyan	Elahe

Office Hours:

Mon:	2:00 -- 3:30pm, Sherrerd 006,	Yuyan
Tue:	11:00 -- 12:30pm, Sherrerd 006,	Koushiki
Wed:	3:00 -- 4:30pm, Sherrerd 006,	Carson
Thu:	10:00 -- 11:30am, Sherrerd 006,	Elahe
Thu:	3:00 -- 4:30pm, Sherrerd 006,	Huanran
Fri:	1:30 -- 3:00pm, Sherrerd 209,	Prof. Vanderbei

Old Midterms and Final Exams: click here

Homework:	45%
Midterm 1:	15%
Midterm 2:	15%
Final:	20%
Participation:	5%

ORF 245: Statistics Fall '15 (Click here for Fall '14) Lectures Based on Deveaux's Forthcoming Book