3.4.2 Standard Normal Distribution
3.4.2 Standard Normal Distribution¶
Overview
This section focuses on the standard normal distribution, a special case of the normal distribution where the mean (μ) is 0 and the standard deviation (σ) is 1. This distribution is crucial in statistical analysis due to its role in simplifying calculations and understanding the properties of any normal distribution.
Key Concepts
- Standard Normal Distribution: Describes a normal distribution with a mean of 0 and a standard deviation of 1, often denoted as Z.
- Critical Values (Z-Scores): These are points on the Z scale corresponding to specified tail areas; critical values are used extensively in hypothesis testing and setting confidence intervals.
Detailed Explanation
Properties of the Standard Normal Distribution
- Symmetry and Shape: The distribution is perfectly symmetrical around zero, exhibiting a bell-shaped curve. The total area under the curve equals 1.
- Z-Scores: These scores measure the number of standard deviations an element is from the mean. A Z-score of 1.0 indicates a value one standard deviation above the mean.
Critical Values and Their Use
- Definition: Critical values (Z-scores) such as Zα/2 are used to determine the cutoff points on the standard normal curve for a given confidence level.
- Examples of Critical Values:
- Z0.05 = 1.645 (used for 90% confidence level)
- Z0.025 = 1.96 (used for 95% confidence level)
- Z0.005 = 2.576 (used for 99% confidence level)
Computational Methods
- Finding Probabilities: Using standard normal tables or software functions like NORM.S.DIST in Excel to find the probability associated with specific Z-scores.
- Inverse Calculations: Using NORM.S.INV in Excel to find the Z-score corresponding to a given cumulative probability.
Practical Application
- Statistical Inference: Critical values from the standard normal distribution are used to determine whether to reject null hypotheses in significance testing.
Example Calculations:
- Finding the probability of Z < 1.645.
- Calculating the cumulative probability for Z = -1.96.
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