Hypothesis Testing Key Concepts and Errors

Hypothesis Testing Overview

Hypothesis testing is a statistical method used to determine if there is enough evidence to reject a null hypothesis based on sample data.

Confidence Levels

A common confidence level used in hypothesis testing is 95%. This means that if the procedure were repeated many times, approximately 95% of the confidence intervals constructed would contain the true population parameter.

Definitions

  • Null Hypothesis (H₀): A statement asserting that there is no change or difference, the status quo.

  • Alternative Hypothesis (H₁): The claim or statement for which we are trying to find evidence. This typically suggests a change or a difference.

Statistical Notation

  • X: Sample mean

  • S: Sample standard deviation

  • Z: Z-score for the confidence level used in hypothesis testing

  • Interval: The range within which we expect the population mean (μ) to fall with a certain level of confidence. If the sample mean (X̄) falls within this interval, H₀ cannot be rejected, supporting the idea that the sample mean is close to the population mean.

Method of Hypothesis Testing

  1. Formulate hypotheses: decide on the null (H₀) and alternative (H₁) hypotheses.

  2. Collect sample data to calculate the sample mean and standard deviation.

  3. Determine whether the sample evidence is sufficient to reject the null hypothesis.

Example of Hypotheses

  • Null Hypothesis (H₀): μ = 25

  • Alternative Hypotheses:

    • H₁: μ < 25

    • H₁: μ > 25

    • H₁: μ ≠ 25

Type I and Type II Errors

  • Type I Error: Rejecting the null hypothesis when it is actually true. In a judicial context, this would mean declaring someone guilty when they are actually not guilty (i.e., the null hypothesis of innocence is rejected).

    • The probability of making a Type I error is denoted by α.

    • Example: Concluding that more than 2% of children have headaches from a new antibiotic when it is actually less than 2%.

  • Type II Error: Not rejecting the null hypothesis when the alternative hypothesis is true. This would mean concluding someone is not guilty when they actually are guilty.

    • The probability of making a Type II error is denoted by β.

    • Example: Concluding that only 2% of children have headaches from an antibiotic when in fact more than 2% do.

Practical Application Case Study

Medco Pharmaceutical Company Scenario

  • Context: Researchers from the FDA are testing whether the percentage of children experiencing headaches from a new antibiotic is greater than 2%.

  • Hypotheses formulated:

    • Null Hypothesis (H₀): p = 0.02 (2% experience headaches)

    • Alternative Hypothesis (H₁): p > 0.02 (more than 2% experience headaches)

  • Type I Error: Incorrectly concluding that more than 2% of children have headaches when the true percentage is actually less than 2%.

  • Type II Error: Failing to reject the null hypothesis when in fact more than 2% of children experience headaches from the antibiotic.

Conclusion Statement After Hypothesis Testing

  1. If the null hypothesis is rejected: There is enough evidence to support the claim that the alternative hypothesis is true.

  2. If the null hypothesis is not rejected: There is not enough evidence to support the claim for the alternative hypothesis.