Statistics and Z-Distribution

Normal Distribution Notation

• Notation: Normal distributions are often represented as N(μ, σ²) where:
  • N indicates normal distribution
  • μ is the mean of the distribution
  • σ² is the variance (standard deviation squared)

Z Distribution

• Z Score Definition: It measures how many standard deviations an observed value (x) is from the mean (μ).
• Formula: ( Z = \frac{(X - μ)}{σ} )
• Example: If the mean is 7.5 and standard deviation is 2.2, with an observed value of 6:
  • Z Score Calculation: ( Z = \frac{(6 - 7.5)}{2.2} ) gives approximately 1.136.

Characteristics of the Normal Curve

• Shape and Properties:
  • The normal distribution is symmetric and bell-shaped.
  • The height of the curve at a particular value represents the probability of that value occurring.
  • For any given normal distribution, repeating the sampling process produces consistent statistics due to the theoretical nature of the normal curve.

Randomization Distribution

• Definitions: Randomization distribution refers to the distribution of a statistic computed from random samples.
  • As sample size increases, this distribution approaches normality.
• Comparison to Theoretical Distribution: Randomization distributions may differ slightly from theoretical distributions, reflecting sample variability.

Statistical Tests

• Hypothesis Testing: Determining if there is enough evidence to support an alternative hypothesis over a null hypothesis.
  • Example: Testing if a difference in proportions is significant.
• Standard Error: The standard error is essential for calculating Z scores and should be derived from the randomization distribution metrics.

Excel for Calculating Statistics

• Using Excel for Calculations:
  • Recommended to create templates in Excel that can automatically perform calculations by inputting relevant statistics.
  • Example: To calculate a Z statistic, use:
  • Observed Statistic (like sample mean)
  • Null Parameter (e.g., expected population mean)
  • Standard Error
  • Sample Formula: ( Z = \frac{(observed - null)}{standard error} )

Examples and Implications

• Case Study Example: If a study reports a proportion difference of 15% between two groups of mosquitoes (one infected and one not), calculate the Z score to assess significance.
  • Observation: A Z score may indicate the likelihood of observed differences under the null hypothesis.
• Establishing Null and Alternative Hypothesis: Carefully define hypotheses to prevent misunderstanding in statistics.
  • Null Hypothesis (H0): For example, ( μ = 10 ) (population mean hours).
  • Alternative Hypothesis (H1): e.g., ( μ < 10 ) (population mean hours less than indicated).

Summary of Methods for Beta Calculation

• Beta (type II error probability assessment) can be calculated utilizing three approaches:
  • Randomization Distribution through simulation
  • Approximating with a smooth theoretical curve centered around the null hypothesis
  • Standardizing the statistics using the Z score for comparison with a standard normal distribution.
• The approaches help validate findings across different scenarios in statistical analysis.

Practical Applications

• Use a consistent alpha level (commonly 0.05) when evaluating statistical significance.
• Building Excel programs streamlines repetitive calculations for multiple hypothesis tests.