The Standard Normal Distribution

Lesson Goals

  • Determine area under the normal curve using tables.

Understanding Normal Distribution

  • Property: Area under any part of the normal curve equals the probability of the random variable falling within that region.

  • Area to the left of a specific value x:

    • Represents the probability that the random variable X is less than or equal to x (P(X ≤ x)).

  • Area to the right of x:

    • Represents the probability that the random variable X is greater than or equal to x (P(X ≥ x)).

Key Concepts

  • Continuous Probability: For a continuous distribution, P(X ≤ x) is equivalent to P(X < x).

  • Calculating Area: Can be accomplished using calculus, but it is not covered in this lesson.

Standard Normal Distribution Tables

  • There are several types of standard normal distribution tables, focus will be on:

    • Cumulative standard normal distribution table.

  • Values in the table are rounded to two decimal places:

    • First decimal place: Listed in the left column.

    • Second decimal place: Listed in the top row.

    • Intersection gives the area under the curve to the left of that z value.

Finding Areas

Area to the Left of a Z Value

  • Look up the z value in the cumulative table.

Area to the Right of a Z Value

  • Two methods:

    1. Using Total Area:

      • Area to the right = 1 - Area to the left.

    2. Using Symmetry:

      • Area to the right = Area to the left of -z.

Area Between Two Z Values

  • Look up both z values:

    • Subtract the smaller area from the larger area.

Finding Area in the Tails

  • Two methods:

    1. Finding Areas of Both Tails:

      • Area in left tail = area to the left of first z value.

      • Area in right tail = area to the right of second z value (use complementary or symmetry).

      • Total area = Area in left tail + Area in right tail.

    2. Using the Area Between Z Values:

      • Find area between z values, subtract from one.

Summary of Procedures

  1. Area to the Left of Z Value: Look up z value in the cumulative table.

  2. Area to the Right of Z Value: Use negative z value in the cumulative table.

  3. Area Between Two Z Values: Look up both, subtract smaller from larger.

  4. Area in the Tails for Two Z Values: Find both areas and add together.