Lecture Notes: Handling Unknown Population Standard Deviation (Sigma)
Introduction
- Key Question: What if we don't know the population standard deviation (Sigma)?
- Context: We previously assumed knowledge of mu (mean), which is unrealistic.
- Sigma: Represents the standard deviation of the population.
Transition to Sample Standard Deviation
- Problem: Assuming knowledge of Sigma is often unrealistic.
- Solution: Use the standard deviation of the sample instead.
- Notation: Sample standard deviation is denoted as "s."
Implications of Using Sample Standard Deviation
- Good News:
- We can use "s" for calculations.
- Bad News:
- Distribution is no longer normal.
- We must use the Student's T distribution instead.
Student's T Distribution
- Characteristics:
- Similar to the normal distribution.
- Symmetric with a middle value of zero.
- Visually similar to the normal distribution.
- Variation:
- Changes based on sample size (n).
- Requires degrees of freedom (DF), calculated as
n - 1.
Commands and Calculations
- Excel Commands for T Distribution:
- Normal Distribution Command:
NORM.DIST
- T Distribution Command:
T.DIST
- Parameters Needed:
- X (really T value)
- Degrees of Freedom
- Number of Tails (1 or 2)
Example in Excel
- T Value: 0.5
- Degrees of Freedom: 29
- Tails: 1 tail
Visual Representation
- T-Distribution Drawing:
- Similar look to normal distribution.
Using Stat Disk for T Distribution
- Process:
- Go to
Analysis > Distribution.
- Select
Student T Distribution.
- Input degrees of freedom and T value.
- Click
Evaluate to see the area calculations.
- Provides area to the left, right, two tails, and central area.
Summary
- Excel T Distribution Command:
T.DIST(X, Degrees of Freedom, Tails)
- Options for Tails:
- Can specify either 1 or 2 tails.
These notes cover the transition from assuming known population standard deviation to using sample standard deviation and the implications on statistical distribution, particularly the use of the Student's T distribution with Excel and Stat Disk.