Lesson 13: Estimation with Confidence Intervals

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]

Assignment:

• 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.

Study:

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

Lesson 12: Sampling Distributions and the Central Limit Theorem

Review:

• Exam 2 Results

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

Assignment:

• Complete Exercises 5.36, 5.37, 5.38, 5.39 on pp. 336-340
• Read pp. 335-344, Sampling Distribution of a Sample Mean

Exam 2 Results

Review for Exam 2

Exam 2 on Wed, Oct 11

• Open book, notes, calculator
• No phones, laptops, watches
• No talking with neighbors
• No leaving classroom

Topics:

• Scatterplots
• Linear Regression
  • Sum of Squares
  • Regression line calculations
  • Pearson Correlation Coefficient
  • R-Squared
• Design of Experiments
  • Explanatory variables
  • Response variables
  • Subjects
  • Treatments
  • Placebo effect
  • Bias
  • Double-blind
• Sampling
  • Census vs Sampling
  • Sampling Types
    • Voluntary response
    • Simple random sample (SRS)
    • Stratified random sample
• Two-way tables
  • Marginal distribution
  • Joint distribution
  • Conditional distribution
• Causation
  • vs Correlation
  • Lurking variables
  • Retrospective studies
  • Prospective studies
• Probability
  • Random phenomenon
  • Short term outcomes uncertain
  • Long term outcomes predictable
  • Independent trials
  • Discrete vs Continuous
  • Probability Rules
  • Discrete probability distribution table calculations

Lesson 11: Random Variables

Review:

• Probability
• Exam 2 on Wed, Oct 11

Presentation:

• Random Variables
  • Discrete Random Variables
    • Probability distribution table
  • Continuous Random Variables
    • Area under a density curve
  • video

• Example Problem
  • The random variable x, defined below, gives the average grade of 12th grade students in U.S. high schools. The probability distribution for x is given in Table 20.11.
    x = {4 if A average, 3 if B average, 2 if C average, 1 if D average}

    

    a. Find P x( 3) ≥ , the probability that a randomly selected student has a B or better average.
    b. Find P x( 3) < , the probability that a randomly selected student has a below B average. How is this probability related to your answer to (a)?
    c. Make a probability histogram for the distribution of x. What does your graphic display tell you about the distribution of average grades?

Assignment:

Problem 1. The U.S. government collects data on many variables having to do with households.
Let x = the number of children under 15 in a household. The probability distribution for x is shown in the table below.

a. What is the probability that a randomly selected household has at least one child under 15?
b. What is the probability that a randomly selected household has between two and four children under 15? In other words, find P (2 ≤ x ≤ 4).
c. Draw a probability histogram that represents the probability distribution shown in the table above.

Problem 2. Assume that the distribution of weight for 7½-week old hens is normally distributed with mean 544 grams and standard deviation 49 grams. Let w = weight of a randomly selected hen.
a. Sketch normal density curve representing the distribution of w.

Use technology or the standard normal table to find the probabilities in (b) – (d). On a copy of the normal density curve that you sketched for (a), shade the area under the curve that represents each probability.
b. P ( w < 500)
c. P (w ≥ 580)
d. P (500 ≤ w ≤ 580)

Problems 1 and 2 are from AgainstAllOdds_StudentGuide_Unit20 duplicated here for convenience.

Lesson 10: Intro to Probability and Probability Models

Review:

• Two-way tables
• Causation
• Exam 2 on Wed, Oct 11

Presentation:

• You already know basic probability
  • Flip a coin
  • Roll dice
  • Play cards
  • Events, Outcomes, Sample Space

• 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

Assignment:

• Use Your Knowledge Problems 4.17 and 4.18 on p. 252
• Exercise 4.21 on p. 255
• Exercise 4.31 and 4.33 on p. 257

Lesson 9: Two-Way Tables and Causation

Review:

• Design of Experiments
• Census and Sampling
• Samples and Surveys
• Oops, I went out of order
• Late submissions for Lesson 1-5 have been graded
• Colorado Honda market size

Presentation:

• Two-Way Tables
  • Categorical data/variables
    • Nominal (e.g., gender, eye color, race, etc)
    • Ordinal (i.e., ranking or sequence; e.g., 1st, 2nd, 3rd or freshman, sophomore, junior, senior)
  • Marginal distribution = (row or column total)/(grand total)
  • Joint distribution = (cell entry)/(grand total)
  • Conditional distribution = (cell entry)/(row or column total)
  • Example Two-Way Tables (student paid/unpaid work hours per week):

  • 

  • Marginal distribution example: % students who work more than 30 hours = 6/50 = 12.0%
  • Joint distribution example: % of all students who are female and don’t work = 10/50 = 20.0%
  • Conditional distribution example: % of male students who work 11-20 hours = 7/21 = 33.3%
  • video

• The Question of Causation
  • Correlation does not imply causation
  • Spurious Correlations
  • Lurking variables = explanatory variables not (yet) included in analysis
  • Retrospective study = looking back to find possible causes for an established outcome among a sample population
  • Prospective studies = following a sample population over time and studying behaviors possibly linked to likelihood of an outcome
  • video

Assignment:

Refer to the table below for questions 1-3.

1. Add a row to the bottom and a column to the right-end of your table. Compute the marginal totals and enter them into your table.
2. What percentage of the students who answered both questions were male? Female? Show your calculations.
3. What percentage of the students rated their intelligence as Above Average?

4. Members of a high school language club believe that study of a foreign language improves a student’s command of English. From school records, they obtain the scores on an English achievement test given to all seniors. The average score of seniors who had studied a foreign language for at least two years is much higher than the average score of seniors who studied no foreign language. The club’s advisor says that these data are not good evidence that language study strengthens English skills. Explain what lurking variables prevent the conclusion that language study improves students’ English scores.

5. Recent studies have shown that earlier reports seriously underestimated the health risks (such as heart disease) associated with being overweight. The error was caused by overlooking important lurking variables. In particular, smoking tends both to reduce weight and to lead to health problems such as heart disease. (a) Describe how you would do a retrospective study of the link between being overweight and having heart problems. (b) Describe how you would do a prospective study of this same link.

All questions copied for convenience from the Against All Odds Unit 13 & 14 student guides. 

Study:

• Read AgainstAllOdds_StudentGuide_Unit13
• Read AgainstAllOdds_StudentGuide_Unit14

Lesson 8: Design of Experiments, Sampling and Surveys

Review:

• Linear Regression
• Correlation

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
    • Study hours
    • Homework completion
  • 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

• 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

• Samples and Surveys
  • 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
  • video

Assignment:

An automotive parts manufacturer is planning to launch a new product but it only works on Honda vehicles. The company has hired you to estimate the size of the market for their new product in Colorado. You are provided the following data, including the population (in thousands) and the total number of registered vehicles (in thousands) for the 10 largest counties in the State.

• Part A. Use linear regression to estimate the total vehicle population in Colorado by generating a regression equation to estimate Vehicles (y) using Population (x). What additional information will you need to estimate the vehicle population for all of Colorado?
• Part B. Design a survey to estimate the percentage of Honda vehicles in Colorado. What are some possible sources of bias and variability in estimating % Honda ownership?
• Part C. Use your results in Part 1 and Part 2 to generate a numerical estimate of the size of the Colorado market. Explain how you calculated your estimate and what, if any, assumptions were made.

Study:

• Read Units 15, 16, 17
  • AgainstAllOdds_StudentGuide_Unit15
  • AgainstAllOdds_StudentGuide_Unit16
  • AgainstAllOdds_StudentGuide_Unit17

Lesson 7: Linear Regression and Correlation

Review:

• Scatterplots
• Sum of Squares
• Assignments on Slack
  • #latelesson1 channel for Lesson 1-5
  • Use #lesson06 for Monday, use #lesson07 for today
  • Don’t use direct messages or “Slackbot” to post
  • For help, send direct message to Preston: @pdeherrera

Presentation:

• Linear regression (simple, bivariate)
  • Video
  • Calculate equation of the regression line
    • y-hat = b1*x + bo
      • b1 = “slope” of the line
      • b0 = “y-intercept”
    • b1 = SSxy/SSxx
    • b0 = (∑y/n) – b1*(∑x/n)
  • Example
    • Beer party data: {(60,10), (70,12), (80,20), (90,40)}
    • Calculate slope (b1) and y-intercept (b0) for linear equation

• Correlation
  • Video
  • Pearson Correlation Coefficient
    • r = SSxy/√(SSxx*SSyy)
  • R-Squared =(r)^2
  • Example
    • Beer party data: {(60,10), (70,12), (80,20), (90,40)}
    • Calculate Pearson Correlation Coefficient (r)
    • Calculate R-Squared

Assignment:

• Use this simple data set: {(2,12), (4,9), (7,6), (11,3)}
  • Calculate the slope and y-intercept for the regression line.
  • Calculate the Pearson Correlation Coefficient
  • Calculate R-Squared
• Use data from Lesson 6, Problems 1 and 2.
  • Calculate the slope and y-intercept for the regression line.
  • Calculate the Pearson Correlation Coefficient
  • Calculate R-Squared

Study:

• Read Ch. 2.2., pp. 101-105, Correlation
• Read Ch. 2.3., pp. 108-115, Least Squares Regression
• Read AgainstAllOdds_StudentGuide_Unit11
• Read AgainstAllOdds_StudentGuide_Unit12

Lesson 6: Scatterplots and Least Squares Regression

Review:

• Exam 1 Results
• Exam 1 Solutions

Presentation:

• Scatterplots
  • Video
  • Examples
    • Colorado TCAP Scores and % Free-Reduced Lunch
    • Gender Wage Gap
  • Alternative Uses
    • Predicted vs Actual
    • Hotness vs Craziness
• Linear Regression
  • Find a line of best fit for a set of points
  • Least Squares Regression Proof: MethodLeastSquares
  • Example: Beer Party
    • Use temperature to predict 12-packs consumed
    • Data from previous parties (Temp degrees F, 12-packs consumed)
      • Party 1: 60°, 10 12-packs
      • Party 2: 70°, 12 12-packs
      • Party 3: 80°, 20 12-packs
      • Party 4: 90°, 40 12-packs
    • Scatter Plot
    • Table setup
  • Sum of Squares calculations
    • SSxx = ∑x² – (∑x)²/n
    • SSyy = ∑y² – (∑y)²/n
    • SSxy = ∑xy – (∑x*∑y)/n
    • Demonstrate Calculations
      • Beer party data: {(60,10), (70,12), (80,20), (90,40)}
      • Another example: {(2,3), (4,6), (7,9), (11,12)}

Assignment:

Problem 1. A study was conducted on mercury (Hg) concentrations in fish taken from Lake Natoma in California. The researchers were concerned that mercury concentration levels in sample fish tissue might differ depending on the lab testing the fish. Fish tissue samples from 10 largemouth bass were sent to two labs, Columbia Environmental Research Center (CERC) and University of California, Davis (UC Davis), for inter-laboratory comparison. Mercury concentration is measured in micrograms of mercury per gram of fish tissue (dry weight). Use the mercury concentration data below to create a Scatterplot and Calculate Sum of Squares (SSxx, SSyy, SSxy).

Problem 2. Satellites are one of the many tools used for predicting flash floods, heavy rainfall, and large amounts of snow. Geostationary Operational Environmental Satellites (GOES) collect data on cloud top brightness temperatures (measured in degrees Kelvin (°K)). It turns out that colder cloud temperatures are associated with higher and thicker clouds, which could be associated with heavier precipitation. Data consisting of cloud top temperature measured by a GOES satellite and rainfall rate measured by ground radar appear in the table below. Because ground radar can be limited by location and obstructions, having an alternative for predicting the rainfall rates can be useful. Use the temperature and radar rain rate data below to create a Scatterplot and Calculate Sum of Squares (SSxx, SSyy, SSxy).

Both problems copied from the Against All Odds video series guide for Unit 10.

Study:

• Read Ch. 2, pp. 83-92
• Read AgainstAllOdds_StudentGuide_Unit10