Lesson 15: The t-distribution for Small Sample Inference

October 24, 2018

Review:

• Estimation with Confidence Intervals

Presentation:

• t-distribution
  • For larger samples, n ≥ 30 use Z-table
  • For small samples, n < 30 use t-distribution
  • Only impacts calculation of the test statistic
    • margin of error for confidence intervals
    • test statistic and p-value for significance testing (don’t worry, we’ll cover  these topics later)
  • t distribution critical values
    • print the t distribution table for the next exam
      • use the “0.025” column for estimation with 95% confidence intervals
    • select the row corresponding to df = n – 1
      • df = degrees of freedom
      • n = number of records in sample data
    • find the t* critical value
      • row = df = n – 1
      • column = .025 (top label) or 95% confidence level (bottom label)
      • e.g., if n = 13
        • df = n – 1 = 12
        • go to the row where df = 12 and move to the .025 column
        • you should find t* = 2.179
  • t-statistic formulas
    • for confidence intervals
      • x-bar ± t * s/√n
      • instead of the Z equivalent:  x-bar ± z * σ/√n
  • Video
  • Examples:
    • Identify appropriate t distribution critical values
      • n=10 with 95% confidence, t* = 2.262
      • n=20 with 95% confidence, t* = 2.093
    • A sample of size n = 20 produced the sample mean, x-bar = 36.0 and standard deviation, s = 9.0. Construct a 95% confidence interval to estimate the population mean.
      • n = 25, x-bar = 36.0, s = 9.0
      • df = 24, t* = 2.064
      • m = 2.064*(9/(√25)) = 3.7152
      • Estimate for μ = 36.0 ± 3.7152
      • 95% confidence interval: [32.2848, 39.7152]
        • previous lesson interval with Z=1.96 and n=400: [35.118, 36.882]
        • sample size can make a big difference

Activity:

• Problem 1. Use the t distribution table, assuming 95% confidence, to find the value of t* for each of the following sample sizes.
  • n = 12
  • n = 18
  • n = 24
  • n = 30

• Problem 2. The weights (in lbs) from a random sample of n=16 four-year-old children who took part in a study on childhood obesity are provided below. Estimate the mean weight of four-year-old children using a 95% confidence interval.
  • {37.1, 26.7, 36.1, 36.2, 40.3, 43.9, 36.2, 40.7, 42.5, 34.8, 37.9, 34.5, 31.1, 36.4, 35.7, 33.4}

• Problem 3. After purchasing a new fuel-efficient vehicle, the owner calculates miles per gallon (mpg) the first n=12 times he fills the fuel tank. His calculations are provided below. Estimate vehicle fuel efficiency using a 95% confidence interval.
  • {42, 43, 37, 36, 34, 45, 48, 43, 38, 42, 43, 46}

These examples and problems are from the Against All Odds video series and Introduction to the Practice of Statistics (6th Ed.) by Moore et al.

Assignment:

• In Sheets, use 2014 voter turnout percentages and current voter registration data to estimate voter turnout in your assigned county. Provide a point estimate with a 95% confidence interval. [Yes, you already estimated voter turnout using linear regression. This is an alternative approach.]