Statistics and Z-Distribution

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 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.

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.

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.

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.

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} )

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).

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.

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