Chapter 9: Fundamentals of Hypothesis Testing: One-Sample Tests

Chapter 9: Fundamentals of Hypothesis Testing: One-Sample Tests

Contents

  • 9.1 Fundamentals of Hypothesis Testing, and Z-Test of Hypothesis for the Mean (σ known)
  • 9.2 t-Test of Hypothesis for the Mean (σ unknown)
  • 9.3 One-Tail Tests
  • 9.4 Z-Test of Hypothesis for the Proportion

9.1 Fundamentals of Hypothesis Testing

  • Definition of Hypothesis: A claim or assertion about a population parameter.
Types of Hypotheses

  • Null Hypothesis (H0):

    • States the status quo, the claim to be tested.
    • Contains a statement of equality.
    • Concerns the population parameter (e.g., mean µ, proportion π), not a sample statistic (e.g., sample mean, sample proportion).

  • Alternative Hypothesis (H1):

    • Contradicts the null hypothesis.
    • Contains statements indicating “not equal to” (≠), “less than” (

  • Mutual Exclusivity: H0 and H1 are always mutually exclusive; only one can be true.

Forms of Hypotheses

  • One-Sided Hypotheses:

    • Left-Tail (Lower-Tail) Test:
    • H0: θ ≥ θ0
    • H1: θ < θ0
    • Right-Tail (Upper-Tail) Test:
    • H0: θ ≤ θ0
    • H1: θ > θ0

  • Two-Sided Hypothesis:

    • H0: θ = θ0
    • H1: θ ≠ θ0
Example (9.1): Oxford Cereals
  • Scenario: As plant operations manager, the goal is to monitor cereal box weights.
  • Specification: Mean weight must be 368 grams per box.
  • Action: Sample 25 boxes to compare the mean weight against 368 grams.
  • State Hypotheses:
    • H0: µ = 368 grams
    • H1: µ ≠ 368 grams
Process of Hypothesis Testing
  1. Sample the Population and find the sample mean.
  2. Decision Rule: If the sample mean significantly deviates from the hypothesized mean, then reject H0.

9.2 Z-Test of Hypothesis for the Mean (σ known)

  • Test Statistic: Convert sample statistic to the Z statistic:
    Z<em>STAT=Xˉ−μ</em>0σ/nZ<em>{STAT} = \frac{\bar{X} - \mu</em>0}{\sigma / \sqrt{n}}
  • Critical Value Approach: Establish critical Z values (±Zα/2). If Z_{STAT} falls in the rejection region (> Zα/2 or < -Zα/2), reject H0.
Steps in the Z-Test
  1. State Hypotheses (H0, H1)
  2. Choose Level of Significance (α)
  3. Determine Critical Values to divide rejection and nonrejection regions.
  4. Collect Data, compute test statistic.
  5. Make Decision and interpret results.
Example of Z-Test (9.2)
  • Situation: A restaurant manager measures if the mean waiting time to place an order has changed from 4.5 minutes.
  • Parameters:
    • Sample Size (n) = 36;
    • Hypothesized Mean (µ0) = 4.5
    • Sample Mean (X̄) = 5.1;
    • Standard Deviation (σ) = 1.2 minutes.
  • Calculate test statistic and compare with critical values to reach a conclusion about H0.

9.3 One-Tail Tests

  • Forms of Hypotheses:
    • Lower Tail: H0: θ ≥ θ0; H1: θ < θ0
    • Upper Tail: H0: θ ≤ θ0; H1: θ > θ0
  • Critical Values:
    • One critical value for one-tailed test; negative for left-tailed or positive for right-tailed tests.
Example of One-Tail Test (9.5)
  • Hypothesis Testing Claim: Mean cell phone bill > $52.
  • Parameters:
    • H0: µ ≤ 52
    • H1: µ > 52
    • Conduct t-Test with sample results to determine if there is evidence to support H1.

9.4 Z-Test of Hypothesis for the Proportion π

  • Definition: Deals with categorical variables having two outcomes (e.g., success/failure).
  • Sample proportion (p) formula:
    p=Xnp = \frac{X}{n} where X is the number of successes in sample size n.
Conditions for Normal Approximation
  • Both $nπ$ and $n(1−π)$ must be at least 5 to use normal approximation for p.
  • Test statistic for proportion:
    Z=p−π<em>0π</em>0(1−π0)nZ = \frac{p - π<em>0}{\sqrt{\frac{π</em>0(1-π_0)}{n}}}
Example of Z-Test for Proportion (9.6)
  • Marketing company claims 8% response from mailing.
  • Sample: 500 surveyed, with 25 positive responses.
  • Hypotheses:
    • H0: π = 0.08
    • H1: π ≠ 0.08
  • Use Z-Test to calculate and compare sample data against claims.
p-Value Approach
  • Definition of p-value: Probability of obtaining a test statistic as extreme as observed if H0 is true.
  • Rejection Rule: If p-value < α, reject H0.
    • Calculate p-value using the test statistic and make statistical decisions based on comparison with α.

Conclusion

  • This chapter outlines critical steps and methods for hypothesis testing, discusses the importance of understanding both Z and t-tests under various conditions, and highlights decision-making processes for null and alternative hypotheses across one-sample tests. This serves as a foundation for conducting thorough hypothesis testing in a statistical context.