Activity 22

MATH 216: Statistical Thinking

Hypothesis Testing Foundations

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. Explain the “presumption of innocence” principle in hypothesis testing and how it relates to the null hypothesis.
  1. What is the difference between Type I and Type II errors? Provide real-world analogies for each.
  1. How does the Central Limit Theorem enable hypothesis testing for means with large samples?

Part 2: Real-World Case Study - Pipe Strength Testing (4 minutes)

Context: City regulations require residential sewer pipes to have an average breaking strength greater than 2,400 pounds per foot. A manufacturer tests 50 pipes and obtains: \(\bar{x} = 2,460\) pounds, \(s = 200\) pounds.

  1. State the hypotheses: \(H_0\): \(H_a\):

  2. Calculate the test statistic: \(z\) =

  3. Critical value (\(\alpha = 0.05\)):

Show your work for the test statistic calculation:

Part 3: Decision Making and Interpretation (3 minutes)

Make decisions and interpret results:

  1. Should we reject \(H_0\)? Why? What is your statistical conclusion?
  1. What are the practical consequences of Type I and Type II errors in this pipe strength scenario?
  1. Critical Thinking: Why is this considered a right-tailed test? What does the rejection region represent in practical terms?