In this section, the calculations involving a normally distributed variable, denoted as , where is the mean and is the variance, will be explored in-depth. This exploration includes calculating probabilities, deriving relationships between variables, and the standardization process.
Basics of Normal Distribution
- Normal distribution is a bell-shaped curve reflecting a continuous probability distribution.
- Symmetry and Mean: The curve is symmetric around the mean, showing that values near the mean are more common.
- Standard Deviation : Determines the distribution's spread. A larger means a wider spread.
- Real-World Examples: Common in heights, test scores, and measurement errors.
Probability Calculations in Normal Distribution
- Calculating Probabilities: Focuses on the area under the curve.
- Standard Normal Distribution Table: Shows the probability of a standard normal variable being within a range. It lists probabilities for values less than a given -score.
Example Problems
Example 1: Finding Probability (P(X > 12))
- Standardize the Variable: Convert to using . . For .
- Calculate Probability: Find P(Z > 1) \approx 0.1587. This means P(X > 12) = 0.1587, or a 15.87% chance will be greater than 12.
![Probability Graph](https://tutorchase-resources-platform-client-production.vercel.app/_next/image?url=https%3A%2F%2Ftutorchase-production.s3.eu-west-2.amazonaws.com%2Fcc16394e-440b-4210-a92b-efe046692cfa-file.jpg&w=1080&q=75)
Example 2: Deriving Value of
- Find the -Score: For P(X < x_1) = 0.8413, -score is about 1.00.
- Calculate : Using , find . There's an 84.13% chance will be less than 23.
![Probability Graph](https://tutorchase-resources-platform-client-production.vercel.app/_next/image?url=https%3A%2F%2Ftutorchase-production.s3.eu-west-2.amazonaws.com%2F136ba3ac-5ad0-4d5f-ba6b-6945631839a1-file.jpg&w=1080&q=75)
Example 3: Standardizing a Normal Variable
- Calculate the -Score: For in , .
- Graphical Representation: Shows is standard deviations above the mean of 30.
![Probability Graph](https://tutorchase-resources-platform-client-production.vercel.app/_next/image?url=https%3A%2F%2Ftutorchase-production.s3.eu-west-2.amazonaws.com%2F4e58dfd1-deef-48cd-bedf-c6035b2f8ddf-file.jpg&w=1080&q=75)