Hypothesis Testing Problems and Error Analysis

Q1. Meditation and Stress Levels

  • Problem: Testing if meditation lowers stress levels.
  • Background: Average stress score is 42.
  • Sample Data: 87 people meditated, average score 38, standard deviation 25.
  • Hypothesis Test:
    • Null Hypothesis (H_0): Meditation has no effect on stress levels.
    • Alternative Hypothesis (H_a): Meditation lowers stress levels.
  • Significance Level: \alpha
  • Sampling Distribution of H_a:
  • Decision: (Reject / Fail to reject) H_0
    • Because p-value
  • Conclusion:
    • There (is / is not) sufficient evidence to say that meditation lowers stress levels.
  • Error Analysis:
    • Possible Type I or Type II error?
    • Consequences of the error.

Errors in Hypothesis Testing (Question 1 Context)

  • Type I Error: Rejecting the null hypothesis when it is true (false positive).
  • Type II Error: Failing to reject the null hypothesis when it is false (false negative).

Q2. Honey vs. Conventional Salves for Wound Infections

  • Problem: Comparing honey to conventional salves in treating wound infections.
  • Background: 9\% of wounds treated with conventional salves get infected.
  • Sample Data: 150 wounds treated with honey, 29 became infected.
  • Hypothesis Test:
    • Null Hypothesis (H_0): Honey's infection rate is the same as conventional salves (9%).
    • Alternative Hypothesis (H_1): Honey's infection rate is different from conventional salves.
  • Significance Level: \alpha
  • Sampling Distribution of H_1:
  • Decision: (Reject / Fail to reject) H_1
    • Because p-value…
  • Conclusion:
    • There (is / is not) sufficient evidence to say that honey's infection rate is different.

Effects of Changing Parameters on Error Probabilities (Question 2 Context)

  • Sample Size:
    • Increasing sample size DECREASES the probability of committing a Type I Error.
  • Alpha (Significance Level):
    • Decreasing alpha INCREASES the probability of committing a Type II Error.
  • Distance Between H_0$$ and True Population:
    • Increasing distance INCREASES the probability of committing a Type I Error.