Activity 20

MATH 216: Statistical Thinking

t-Distributions and Confidence Intervals

Time Allocation: 15 minutes total (5 min reading, 10 min individual work)

Part 1: Conceptual Understanding (3 minutes)

Instructions: Answer the following questions:

  1. When should we use the t-distribution instead of the normal distribution for confidence intervals, and why?
  1. How do degrees of freedom affect the shape of the t-distribution, and what happens as sample size increases?
  1. What are the key differences between t-critical values and z-critical values for the same confidence level?

Part 2: Confidence Interval Calculations (4 minutes)

Calculate confidence intervals:

  1. Medical study: \(\bar{x} = 25\), \(s = 4\), \(n = 12\) 95% CI =

  2. Small sample CI: \(\bar{x} = 50\), \(s = 8\), \(n = 10\) 95% CI =

  3. Blood pressure: \(\bar{x} = 2.62\), \(s = 0.95\), \(n = 6\) 95% CI =

Calculate critical values:

  1. Find t-critical values:

    • 95% CI, df=8: \(t^*\) =
    • 99% CI, df=15: \(t^*\) =
    • 90% CI, df=20: \(t^*\) =

Show your work for one calculation:

Part 3: Interpretation and Application (3 minutes)

Interpret results and answer questions:

  1. For the small sample CI (\(\bar{x} = 50\), \(s = 8\), \(n = 10\)), what does the 95% confidence interval tell us about the population mean?
  1. Compare the t-critical values from part 2 with their z-distribution counterparts. Why are t-values larger?

Critical Thinking: What assumptions must be satisfied when using t-distributions for confidence intervals, and why are these more important for small samples?