Lesson 12: Estimation with a Small Sample Using the t-distribution

October 12, 2026

Review:

• Estimation with Confidence Intervals
• Lesson 11 answer key
• Inference for proportions
• Exam 3 on Wed 21 Oct

Presentation:

• What if the best you can do is a sample size of n<30?
• t-distribution
  • For larger samples, n ≥ 30 use Z-distribution
  • For small samples, n < 30 use t-distribution
  • For large samples, the Z and t distributions produce approximately the same results. For small samples, the difference is important.

  • 

  • Only impacts calculation of the margin of error
    • Changes the critical value used to calculate the margin of error.
    • p-value for significance testing (don’t worry, we cover  these topics later)
  • For small-sample t intervals, we assume the observations are independent and the population distribution is approximately Normal, without strong outliers.
  • t distribution critical values
    • print the t distribution table for the next exam
      • use the 95% confidence or “0.025” column
    • 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 (bottom label) or 95% confidence level (top 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* = 2.18
  • t-statistic formulas
    • for confidence intervals
      • x̅ ± t * s/√n
  • Video – Against All Odds Unit 26
  • Examples:
    • Identify appropriate t distribution critical values
      • n=10 with 95% confidence, t* = 2.26
      • n=20 with 95% confidence, t* = 2.09
    • A sample of size n = 25 produced the sample mean, x̅ = 36.0 and standard deviation, s = 9.0. Construct a 95% confidence interval to estimate the population mean.
      • n = 25, x̅ = 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 = 28
• 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. The sample standard deviation equals 4.2. {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. You will need to calculate standard deviation. {42, 43, 37, 36, 34, 45, 48, 43, 38, 42, 43, 46}
• Problem 4. Two researchers are estimating the same population mean. Both obtain a sample mean of 50 and a sample standard deviation of 10. Researcher A uses n = 10 observations. Researcher B uses n = 25 observations.
  • Which researcher will have the wider 95% confidence interval?
  • Explain before doing any calculations.
  • Estimate the population mean for Researcher A.
  • Estimate the population mean for Researcher B.
• Problem 5. A grocery store manager wants to estimate the average amount customers spend per visit. A random sample of 20 customers produces a sample mean of $64.50 and a sample standard deviation of $18.00.
  • Identify the sample size, sample mean, and sample standard deviation.
  • Determine the degrees of freedom and find the appropriate t* critical value for 95% confidence.
  • Calculate the margin of error.
  • Construct a 95% confidence interval for the population mean.
  • Interpret your confidence interval in plain English.

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.