Activity 26

MATH 216: Statistical Thinking

Comprehensive Hypothesis Testing Framework

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

Part 1: Conceptual Understanding (3 minutes)

Instructions: Answer the following questions based on the lecture content:

  1. What are the three types of hypothesis tests (based on alternative hypothesis direction) and when should each be used?
  1. Explain the difference between the null and alternative hypotheses, and why the null hypothesis always contains equality.
  1. How do degrees of freedom affect the t-distribution and why is this important for small sample inference?

Part 2: Real-World Hypothesis Testing Applications (4 minutes)

Apply comprehensive hypothesis testing framework to real-world scenarios:

  1. Two-Tailed Test - Pharmaceutical Research: A company tests if a new drug changes reaction time (n=12)

    • Sample: \(\bar{x} = 128\) ms, \(s = 15\) ms, test \(H_a: \mu \neq 120\) ms
    • Calculate t-statistic and critical value (\(\alpha\) = 0.05)

    t-statistic = Critical value =

  2. Right-Tailed Test - Manufacturing Improvement: A process claims to increase production rate (n=10)

    • Sample: \(\bar{x} = 52\) units/hour, \(s = 4\) units/hour, test \(H_a: \mu > 50\) units/hour
    • Calculate t-statistic and critical value (\(\alpha\) = 0.05)

    t-statistic = Critical value =

  3. Left-Tailed Test - Environmental Monitoring: Study tests if pollution levels decreased (n=8)

    • Sample: \(\bar{x} = 18\) ppm, \(s = 3\) ppm, test \(H_a: \mu < 20\) ppm
    • Calculate t-statistic and critical value (\(\alpha = 0.05\))

    t-statistic = Critical value =

Show your work for one complete calculation:

Part 3: Decision Making and Interpretation (3 minutes)

Make statistical decisions and interpret results:

  1. For the pharmaceutical research case (t = 1.85, critical = 2.201), what is your statistical conclusion? What does this mean in practical terms?
  1. For the manufacturing improvement case (t = 1.58, critical = 1.833), what is your statistical conclusion? Why might this result occur even if there is a real improvement?
  1. For the environmental monitoring case (t = -1.89, critical = -1.895), what is your statistical conclusion? What are the implications for environmental policy?

Critical Thinking: Why is it important to choose the correct type of hypothesis test (two-tailed vs one-tailed) before collecting data?