Hypothesis Testing (2)

Foundations of Predictive Analytics & Decision Modeling - Hypothesis Testing

• Overview of Hypothesis Testing

  • Assessing claims about population parameters (mean, proportion).

  • Involves formulating null (H0) and alternative hypotheses (H1).

• P-Value Approach

  • P-value indicates the probability of observing the test statistic under H0.

  • Reject H0 if p-value < significance level (𝛼).

  • Can be calculated using statistical tables or Excel.

• Testing a Proportion

  • Involves similar steps to test means.

• Two-Sample Hypothesis Testing

  • Compares means or proportions from two different samples to determine if they are significantly different.

• Example: Soup from Vending Machines

  • Sample data: n = 25, X̄ = 3.97 oz, S = 0.04 oz.

  • A) Hypothesis test at 𝛼 = 0.02:

  • H0: µ = 4 oz, H1: µ ≠ 4 oz

  • Test statistic (T): $T_{24} = \frac{3.97 - 4.0}{0.04/\sqrt{25}} = -3.75$

  • Results: Reject H0 since p < 0.02.

  • B) Construct 98% confidence interval:

  • CI: 3.97 \pm 2.4922(0.008) => [3.95, 3.99]

  • 4.00 oz not included in the CI.

  • C) P-value approach:

  • Critical Z-value = -2.4922; P-value = 0.0128,< 𝛼 = 0.02, so reject H0.

• Example: Hammermill Company

  • Test if mean paper width exceeds specification (216 mm).

  • Sample data: n = 50, X̄ = 216.0070 mm, S = 0.0230.

  • Hypothesis: H0: µ = 216 mm, H1: µ ≠ 216 mm.

  • Decision rule at 𝛼 = 0.05: Reject H0 if Z > 1.96 or Z < -1.96.

• Two-Tail Test Execution

  • State hypotheses, determine decision rule, calculate test statistic, make decision, and take action based on results.

• Example: Reading Scores

  • H0: µ ≥ 70 versus H1: µ < 70.

  • Sample: n = 16, X̄ = 68, S = 9, test results suggest to not reject H0.

• Impurities in Water

  • Company claims at most 1 ppm of benzene; test shows H0 rejected (1.16 ppm observed).

  • Sample data: n = 25, X̄ = 1.16 ppm, S = 0.20 ppm.

  • H0: µ ≤ 1 ppm, H1: µ > 1 ppm.

  • Results indicate contamination likely present.