HYPOTHESIS-TESTING-2

1. Hypothesis Testing Overview

  • Definition: Hypothesis testing is a statistical method that determines if a claim about a population parameter is valid based on sample data.

  • Purpose: To assess the validity of our beliefs or assumptions about a population.

2. Methods for Hypothesis Testing

2.1 Traditional Method

2.2 P-value Method

2.3 Confidence Interval Method

3. Formulation of Hypotheses

3.1 Null Hypothesis (H0)

  • Represents the initial assumption; states no significant effect or difference exists between variables.

  • Example: H0: The new drug has no effect on blood pressure.

3.2 Alternative Hypothesis (HA or H1)

  • Asserts that a difference or effect does exist.

  • Example: H1: The new drug reduces blood pressure.

4. Significance Level (α)

  • Alpha (α): A threshold set to determine how strong evidence is needed before rejecting the null hypothesis.

5. Steps in Hypothesis Testing (Traditional Method)

  1. State Hypotheses: Identify H0 and H1.

  2. Find Critical Values: Determine cutoff points based on α.

  3. Compute Test Statistic: Measure evidence against H0.

  4. Decision: Reject or not reject H0 based on comparison of test statistic to critical values.

  5. Summarize Results: Provide findings.

6. Z-test for a Mean (Traditional Method)

6.1 When to Use

  • Appropriate for large sample sizes (n ≥ 30) and when population variance is known.

6.2 Z-test Formula

  • Compute test statistic using the formula:Z = (X̄ - μ) / (σ / √n)Where:

    • X̄ = sample mean

    • μ = population mean

    • σ = population standard deviation

    • n = sample size

6.3 Example Problem

  • Claim: Mean age of doctors in a hospital system is greater than 46 years.

  • Decision: Reject H0 if Z > 1.645 at α=0.05.

7. Z-test for a Proportion (Traditional Method)

7.1 Requirements

  • Assumptions include a random sample and binomial experiment conditions.

7.2 Example Problem

  • Claim: 17% of young people are obese; conduct test to validate.

8. T-test for a Mean (Traditional Method)

8.1 When to Use

  • Used for small sample sizes (n < 30) and unknown population standard deviation.

8.2 Example Problem

  • Claim: Average number of infections per week in a hospital is 16.3.

9. Chi-Square Test for Variance (Traditional Method)

9.1 When to Use

  • Tests claims about variance or standard deviation.

9.2 Example Problem

  • Claim: Variance in IQ of patients differs from a known population variance.

10. P-value Method in Hypothesis Testing

10.1 Definition

  • P-value: Probability of obtaining a sample statistic at least as extreme as the one observed, given that H0 is true.

10.2 Decision Rule

  • If P-value ≤ α, reject H0.

  • If P-value > α, do not reject H0.

10.3 Example Problem

  • Claim regarding tuition costs at a college.

11. Confidence Intervals Method

11.1 Relationship to Hypothesis Testing

  • CI provides a range of values likely to include the population parameter based on sample data.

  • If the CI does not contain the hypothesized mean, H0 can be rejected.

11.2 Example Problem

  • Claim about the weight of sugar bags; find the CI to assess validity.