Lesson 18: Significance Testing

Mon, Oct 31

Review:

• Inference for Proportions
• Voter estimation assignment – due in 1 week

Presentation:

• Test of Statistical Significance
• Hypothesis Testing
  • Null hypothesis, Ho (p. 375)
  • Alternative hypothesis, Ha (p. 375)
  • P-value (p. 377)
  • z-test for population mean (p. 383)
  • α = 0.05 (same as 95% confidence)
• Example Problem 6.15 (p. 383-384)
  • Ho: μ = 168, Ha: μ ≠ 168
  • x-bar = 173.7, n=71, σ = 27, α = 0.05
• Tests of Significance Video
• Example Problem 6.16 (p. 385)
  • Ho: μ = 450, Ha: μ ≠ 450
  • x-bar = 461, n=500, σ = 100, α = 0.05

Activity:

• Problems
  • Ho: μ = 25, Ha: μ ≠ 25, x-bar = 28.2, n=64, σ = 8.4, α = 0.05
  • Ho: μ = 7.4, Ha: μ ≠ 7.4, x-bar = 6.9, n=24, σ = 1.2, α = 0.05
  • Ho: μ = 1200, Ha: μ ≠ 1200, x-bar = 1215, n=121, σ = 100, α = 0.05

Assignment:

• Read pp. 372-386, Tests of Significance
• Voter estimation exercise – Due in class on Monday Nov 7
  • Estimate total number of votes cast in Pueblo County
    • For President of the United States
    • For US Senate (Bennet vs Glenn)
    • For US Representative to Colorado’s 3rd Congressional District (Tipton vs Schwartz)
  • Estimate % votes cast in Pueblo County
    • For Hillary Clinton
    • For Donald Trump
    • For Bennet
    • For Glenn
    • For Tipton
    • For Schwartz
    • For (vote Yes) Proposition 200
  • Explain your methodology
  • Hint: pueblovotes.com

Lesson 17: Inference for Proportions

Wed, Oct 26

Review:

• Estimation with Confidence Intervals

Presentation:

• How to Estimate a Population Proportion
  • Conduct a Simple Random Sample (SRS) of size n
  • Record the count, x, of “successes”, e.g., # of voters supporting candidate A
  • Calculate the sample proportion, p-hat = x/n
  • If n is sufficiently large (≥30), we can assume p-hat is Normally distributed
  • Estimate of the population proportion mean, μ = p-hat
  • Estimate of the population proportion std dev, σ = √(p*(1-p)/n)
  • Estimate margin of error, m = z*σ  (use z = 1.96 for 95% confidence)
  • Estimate with 95% confidence interval is p-hat ± m
• Example 8.1 on p. 489
  • n = 13,819
  • x = 3,140
  • p-hat = x/n = 3,140/13,819 = 0.227
  • Standard Deviation = √(p*(1-p)/n) = √(0.227*(1-0.227)/13,819) = 0.00356
  • m = z*σ = 1.96*0.00356 = 0.00698
  • p-hat ± m = 0.227 ± 0.00698
  • 95% CI = [0.22002, 0.23398]

Activity:

• Complete Problems 8.1 and 8.2 on p. 490-491
• Election Polling Questions
  • In a survey conducted by Quinnipiac University (October 10-17), 685 likely Colorado voters were asked which candidate for President they plan to vote for in the election. The candidates and corresponding support levels are listed below. Calculate a 95% confidence interval for each candidate.
    • Clinton 45%
    • Trump 37%
    • Johnson 10%
    • Stein 3%
  • In a poll conducted in September by Colorado Mesa University and Rocky Mountain PBS, 540 registered voters were asked whether they favor or oppose Amendment 70 (increasing the Colorado minimum wage) with the following results. Calculate a 95% confidence interval for the “Favor” and “Oppose” proportions.
    • Favor 58%
    • Oppose 36%
    • Undecided 7%

Assignment:

• Read pp. 487-491, Inference for Proportions
• Read two recent FiveThirtyEight.com articles by Nate Silver
  • Why our Model is More Bullish on Trump than Others
  • Where are the Undecided Voters

Lesson 16: Estimation with Confidence Intervals

Mon, Oct 24

Review:

• Sampling Distributions

Presentation:

• How to Estimate a Population Mean (μ)
  • Conduct a Simple Random Sample (SRS) of size n
  • Calculate the sample mean, x-bar = (∑x)/n
  • If n is sufficiently large (≥30), we can assume x-bar is Normally distributed
  • Estimate of the population mean, μ = x-bar (point estimate)
  • Estimate of the population standard deviation, σ = s/(√n)
  • Estimate margin of error, m = z*σ  (use z = 1.96 for 95% confidence)
  • Estimate with confidence interval for μ = x-bar ± m
• Watch video
• Example: A sample of size n = 400 produced the sample mean, x-bar = 36.0 and sample standard deviation, s = 9.0. Construct a 95% confidence interval to estimate the population mean.
  • n = 400
  • x-bar = 36.0
  • s = 9.0
  • m = 1.96*(9/(√400)) = 0.882
  • Estimate for μ = 36.0 ± 0.882
  • 95% confidence interval: [35.118, 36.882]

Activity:

• Problem 1. A sample of size n = 100 produced the sample mean, x-bar = 16.0 and sample standard deviation, s = 3.0. Construct a 95% confidence interval to estimate the population mean.
• Problem 2. An operation manager at a large plant observed 120 workers assembling an electronic component. The average time needed for assembly was 16.2 minutes with a standard deviation of 3.6 minutes. Construct a 95% confidence interval to estimate the mean assembly time.
• Problem 3. A computer technician installs new hard drives on 64 different computers. The average installation time is 42 minutes with a standard deviation of 5 minutes. Construct a 95% confidence interval for the mean installation time.
• Problem 4. A research firm conducted a survey of regular smokers to estimate the average amount spent per week on cigarettes. A sample of 49 regular smokers revealed average spending on cigarettes to be $21.55 with a standard deviation of $5.21. Construct a 95% confidence interval to estimate mean weekly cigarette spending.

Assignment:

• Read pp. 353-362, Introduction to Inference and Estimating with Confidence

Lesson 15: Sampling Distributions and the Central Limit Theorem

Wed, Oct 19

Review:

• Probability

Presentation:

• Sampling Distributions
  • Take multiple Simple Random Samples, sample size = n, and calculate sample mean
  • “Sampling Distribution” is the distribution of sample means
  • Standard deviation of sample means = σ/√n
  • Example 5.18 on p. 338
• Central Limit Theorem
  • Simple random samples of size n from any population with mean μ and finite standard distribution σ. When n is large (typically ≥ 30), the sampling distribution of the sample mean is approximately Normal.
  • Example 5.19 on p. 339
• Video

Activity:

• Complete Exercises 5.36, 5.37, 5.38, 5.39 on pp. 336-340

Assignment:

• Read pp. 335-344, Sampling Distribution of a Sample Mean

Lesson 14: Intro to Probability and Probability Models

Mon, Oct 17

Review:

• Make-up Exams
• Samples, Surveys and Political Forecasting
• Election Forecast Update

Presentation:

• Intro to Probability
  • Topic worthy of its own course
  • Simulation of outcomes
    • Random phenomenon
      • Individual outcomes are uncertain
      • Regular distribution of outcomes in large number of trials
    • Independent trials
    • Repetition – note coin tosses required to reach 0.5 probability (p. 238)
  • Video
• Probability Models
  • Sample Space
    • Discrete vs Continuous
    • All possible Outcomes
  • Probability Rules (see p. 246)
    • Rule 1: Any probability is a number between 0 and 1. Or, mathematically, 0 ≤ P(A) ≤ 1
    • Rule 2: All possible outcomes together must have probability = 1.
    • Rule 3: If events A and B have no outcomes in common, they are disjoint and P(A or B) = P(A) + P(B). This is the addition rule.
    • Rule 4: The complement of an event, A = 1 – P(A)
    • Rule 5: If events A and B are independent then P(A and B) = P(A)*P(B). This is the multiplication rule.
  • Examples:
    • Coin Toss – see Example 4.8 on p. 245
      • Flip a coin 4 times. Event A = toss is “Heads” exactly 2 times. What is P(A)?
      • Step 1: Define the Sample Space (all possible outcomes)
      • Step 2: Identify the set of outcomes that satisfy Event A
      • Step 3: P(A) = Number of Event A outcomes ÷ Number of possible outcomes
    • Roll a pair of dice
      • Define Event A (e.g., roll = 7). What is P(A)?
      • Follow Steps 1-3 above
  • Video
  • Exercise 4.21 on p. 255

Activity:

• Complete Exercises 4.22, 4.24, 4.26 on pp. 255-256

Assignment:

• Read pp. 237-241, Intro to Probability
• Read pp. 242-248, Probability Models
• Read Where the Race Stands with 3 Weeks to Go by Nate Silver

Lesson 13: Samples, Surveys and Political Forecasting

Wed, Oct 12

Review:

• Census and Sampling
• Late exams and make-up corrections graded by Mon, Oct 17

Presentation:

• Samples and Surveys video
• Toward Statistical Inference
  • Population parameters
  • Sample statistics
  • Sampling distribution
  • Bias and Variability
    • to reduce bias, use random sampling
    • to reduce variability, use larger sample sizes (p. 215)
    • variability determines margin of error
    • see illustration on p. 218
  • Simulation
• The Power of Polling and Poll Aggregation in Politics
  • Nate Silver’s 2012 Political Forecasting
  • Among Election Quants Nate Silver Reigns Supreme
  • Guest Appearance on Jon Stewart’s Daily Show (after 2012 election)
  • http://fivethirtyeight.com/

Activity:

• Read and study this 2016 Presidential Election Forecast at fivethirtyeight.com then answer the following questions.
  • Check out the expected margin of victory by state
    • Compare the graphic to “stacked boxplots”
    • What similarities and differences do you observe?
  • Check out the simulation of electoral votes
    • What type of graphic is used to illustrate the simulation results?
    • How would you explain what’s going on in these graphics to a friend?
  • Check out these estimates of the popular vote
    • What type of graphic is used to illustrate the likely popular vote outcome?
    • How would you explain what’s going on in this graphic to a friend?
  • Choose 1 of 2 options:
    • Option 1: Describe what role Colorado will play in the election using data and analysis on this page.
    • Option 2: Identify at least 2 data elements/graphics or other analyses on the page you don’t understand.

Assignment:

• Read pp. 212-220, Toward Statistical Inference
• Complete Exercises 3.83, 3.84, 3.85, 3.86 and 3.87

Lesson 12: Census and Sampling

Mon, Oct 10

Review:

• Exam corrections due now
• Design of Experiments

Presentation:

• Census and Sampling
  • Census = attempt to count entire population
    • US Census every 10 years
    • Most costly non-military federal government operation (except maybe bailing out banks)
  • Sample = gather info from a portion of the population
    • Sample Types
      • Voluntary response sample
        • Complete an optional survey
        • Inherent bias
        • Example: Google Survey
      • Simple random sample
        • select from a population
        • each individual has equal chance of being selected
        • Example: random selection of subset of enrolled students
      • Stratified random sample
        • divide population into groups or strata
        • random sampling, equal chance of selection within each group
        • Example: random selection within major (30% CIS, 35% Mgmt, 35% Econ)
  • Census and Sampling Video

Activity:

• Complete Exercise 3.56 on p. 208
• Identify the population to sample for the following upcoming election issues:
  • Initiative #101 – State Minimum Wage
  • Ballot Measure 200 – Retail Marijuana

Assignment:

• Read pp. 197-203, Sampling Design

Lesson 11: Design of Experiments

Wed, Oct 5

Review:

• Exam 1 Results
• Make-up due Mon, Oct 10

Presentation:

• Design of Experiments
  • Explanatory variables
  • Response variable
  • Subjects – objects or participants in experiment
  • Treatments – experimental conditions
• Example
  • Explanatory variables
    • Exam 1 Part 1 begin day/time
    • Hours of sleep night before exam
  • Response variable:
    • Exam 1 Score
  • Subjects
    • Students
  • Treatments
    • Open book vs Closed book
    • Written vs Multiple Choice
    • Morning vs Afternoon
• Experimental Design Principles
  • Comparison of Treatments
    • Placebo effect
    • Control group
  • Randomize assignments
    • Bias
    • Double-blind
  • Repetition
    • Limit impact of individual observations
    • Confirm hypotheses
• Design of Experiments video

Activity:

• Divide into groups
• Read and discuss Example 3.7 on p. 178
• Read and discuss Exercise 3.19 on p. 192-193
• Design an experiment
  • Response variable = Exam 2 Test Scores (yes, the exam you will take)
  • Subjects = Students in Inferential Statistics
  • Explanatory variables
    • What data elements could be measured or collected?
  • Treatments
    • What treatments could be implemented?
• Submit written description of your groups Experimental Design
  • How will treatments be compared?
  • How can treatments be randomly assigned to subjects?
  • How can the experiment avoid bias?
  • Include team member names

Assignment:

• Read pp. 178-192, Design of Experiments

Exam 1: Results

Mon, Oct 3

If you missed the exam last week, take it NOW!

Exam 1 Results:

• Mean: 72.8
• Std Dev: 29.5
• Median: 84.0
• Max: 100
• Min: 0

Exam Make-up Option:

• Redo sections to correct mistakes
• Earn back up to 50% of points missed
• Must be submitted in writing
  • Include a cover page with instructions identifying and clarifying corrections
  • Include the original exam
• Due in class on Mon, Oct 10

Exam 1: Review

Mon, Sep 26

Review:

• Exam 1 on Wed, Sep 28
• Take-home assignment due at time of exam

Exam 1 Review Topics:

• Types of Data
• Types of Analysis
• Stemplots
• Histograms
• Mean
• Median
• Percentiles
• Boxplots
• Variance
• Standard Deviation
• Density Curves
• Normal Curves
• Standard Normal Distribution
• Z-Scores
• Z-Table
• Normal Curve Problems

Practice Problems:

• Mean and Standard Deviation
  • 1.70-1.74 on p. 50
• Normal Curve Exercises
  • 1.120-1.124 on p. 74
  • 1.126-1.129 on pp. 74-75
  • 1.136-1.137 on p. 75

Keys to success on Exam 1:

1. Complete and Print Part 1 of the exam no later than Tue night. Don’t wait until the last minute and then show up late due to a printer malfunction.
2. Practice calculating standard deviation and completing Normal Curve problems so you can be as efficient as possible with your time during the exam.
3. Be sure you understand what you’ve done. Don’t just go through the motions. This won’t be a multiple choice test.
4. Come to class prepared. Bring calculator, textbook, notes, pencils/pens and don’t forget your Part 1 Report!
5. Don’t get hammered tomorrow night …. c’mon man.
6. Get a good night’s sleep. Don’t pull an all-nighter and expect to be thinking clearly.
7. Eat something nutritious before the exam so you have fuel in the tank.
8. During the exam – stay calm and focus on one problem at a time. If you get stuck ask me a question to help clarify or move on to the next problem.
9. Think positive – visualize yourself handing in the exam and feeling good about your performance.