Calculating Probabilities Under a Normal Curve

Module 3 - Section 3: Calculating Probabilities Under a Normal Curve

By: Rosana Fok

Introduction to Normal Probabilities

• Objective: To find the probability of a value being within 1.5 standard deviations from the mean in a normal distribution.

• Use of Table z:

  • Table z lists various values of z* along with the cumulative area (cumulative proportion) from negative infinity to z.

  • This table is applicable for standard normal distributions (mean = 0, standard deviation = 1).

Table z Overview

• The following segments provide various z values and their corresponding cumulative areas:

  • Values in the table are calculated to four decimal places.

Table z - Areas under the Standard Normal Curve (Extract)

• For instance, for z = 0.0 (mean):

  • Cumulative area = 0.5000

  z	0.00	0.01	0.02	0.03	0.04	0.05	0.06	0.07	0.08	0.09
  0.0	0.5000	0.5040	0.5080	0.5120	0.5160	0.5199	0.5239	0.5279	0.5319	0.5359
  0.1	0.5398	0.5438	0.5478	0.5517	0.5557	0.5596	0.5636	0.5675	0.5714	0.5753
  …	…	…	…	…	…	…	…	…	…	…

Applications of Table z

• Finding specific areas under the standard normal curve for different z scores.

Example Calculations

• To find the areas corresponding to specific z scores:

  • Area to the left of z = 1.47: Check Table z for the value.

  • Area to the left of z = -1.645: Also via Table z.

  • For extreme z values (e.g., z = -6): Use the cumulative probabilities calculated in Table z.

Extended Table z Values

• As z approaches extreme values, cumulative areas approach either 0 or 1:

  • For z = 3.9, cumulative area = 1.0000.

Standardizing Normal Distributions

• Theorem:

  • If a variable y is normally distributed with mean µ and standard deviation ( \sigma ) (i.e., ( y \sim N(\mu, \sigma) )), then the standardized variable z is also normally distributed:

  • ( z \sim N(0, 1) ) can be calculated using the formula:

  $z = \frac{y - \mu}{\sigma}$

Example Problem: Finding Area for a Non-Standard Normal Distribution

• Given: ( y \sim N(100, 5) )

• Need to find the area between 98 and 107:

  • Standardized values:

  • For 98: $z = \frac{98 - 100}{5} = -0.4$

  • For 107: $z = \frac{107 - 100}{5} = 1.4$

  • Use Table z to find:

  • Area up to z = 1.4 is approximately 0.9192.

  • Area up to z = -0.4 is approximately 0.3446.

  • Compute the area between:

  • P(98 < y < 107) = P(z < 1.4) - P(z < -0.4)

  • Calculation yields:

    • $0.9192 - 0.3446 = 0.5746$

Summary Procedure for Finding Normal Proportions

1. State Problem: Define the observed variable y clearly.

2. Standardize y: Convert y into z to form the problem based on a standard normal variable.

   • Example visual representation: Drawing the area under the curve to aid understanding.

3. Use Table z: Look up the cumulative area corresponding to the z values found.

4. Equations for Different Scenarios:

   • For less than a certain value:
     P(z < a*)

   • For greater than a certain value:
     P(z > a*)

   • For values between a and b:
     P(a* < z < b*)

Notation Recap

• Standard deviation: ( \sigma )

• Mean: ( \mu )

• Random variable: y

Conclusion and Acknowledgements

• Thanks for watching this video!