7.01

Module Overview

  • Module 07: Focuses on Mean and Slope in hypothesis testing, particularly using t-tests versus z-tests.

Hypothesis Testing

One-Sample Mean

  • z-Test vs. t-Test:

    • z-Test: Used when the population standard deviation is known.

    • t-Test: Used when the population standard deviation is unknown; relies on the t distribution.

  • Key Similarity: The process of conducting both tests is largely the same, with adjustments for the t-test.

Considerations for t-Test

  • Normality: Check if the original population is approximately Normally distributed.

  • Sample Size: The sample size influences the robustness of the t-test.

    • n ≥ 30: Assume Normality; the sample is considered sufficiently large.

    • 15 ≤ n < 30: Verify approximate Normality without outliers; proceed cautiously.

    • n < 15: Verify approximate Normality without outliers or skewness; proceed with extreme caution.

  • Data Visualization: Use box plots, histograms, or Normal probability plots to assess Normality.

Properties of t Distribution

  • Critical Value: Unlike the standard Normal distribution, the t distribution calculates the area right of the critical value.

  • Visualization: Review how the t distribution differs in appearance from the standard Normal distribution to understand implications for hypothesis testing.

Practical Application: Inflatable Gorilla Factory Example

Scenario

  • Quality Assurance Team: Needs to determine if fans last longer than 35 days with a 95% confidence level.

  • Hypothesis:

    • Null Hypothesis (H0): µ = 35 days

    • Alternative Hypothesis (H1): µ > 35 days

Sample Data

  • Sample Size (n): 18

  • Sample Mean (x̄): 37.9 days

  • Sample Standard Deviation (s): 8.4 days

Test Statistic Calculation

  • Formula: ( t = \frac{x̄ - μ_0}{s/\sqrt{n}} )

  • Plugging in values: ( t = \frac{37.9 - 35}{8.4/\sqrt{18}} \approx 1.465 )

Finding the p-Value

  • Determine p-value for t = 1.465 with degrees of freedom (df) = n - 1 = 17.

  • Calculation Methods: Use a calculator or t distribution table to find ( P(t \geq 1.465) ).

  • Result: p-value ≈ 0.0806.

Conclusion

  • Interpretation: The p-value is greater than the 5% significance level, leading to the conclusion:

    • There is insufficient evidence to reject the null hypothesis (H0) that population mean is equal to 35 days.

    • Therefore, it cannot be concluded that the inflatable gorilla's fan lasts longer than 35 days.