Chapter 6 part 6

Understanding Normal Distribution Problems

  • Identify the type of probability question: less than, greater than, or between.

    • Less than ((<))

    • Greater than ((>))

    • Between ((<\text{value}_1<\text{value}_2))

Step-by-Step Process

  1. Convert Data Value to Z Score

    • Z Score Formula: (Z = \frac{(X - \mu)}{\sigma})

    • Where (X) = data value, (\mu) = mean, and (\sigma) = standard deviation.

    • Example: Convert 20 gallons, with mean = 15 gallons, and std dev = 6 gallons.

    • Calculation: (Z = \frac{(20 - 15)}{6} = 0.8333), rounded to 0.83.

  2. Sketch the Normal Distribution

    • Draw a bell curve.

    • Label axes with standardized scores: -3, -2, -1, 0, 1, 2, 3.

    • Plot the calculated Z score.

    • Shade the appropriate area based on the problem type:

      • Greater than: shade to the right.

      • Less than: shade to the left.

      • Between: shade between two points.

  3. Look Up Z Score in Normal Distribution Table

    • Use the table to find the area to the left of the Z score.

    • Adjust the result based on whether the problem asks for greater than or less than.

    • Example: For Z = 0.83, area to the left = 0.7967.

    • For greater than problem: (P(X>20) = 1 - 0.7967 = 0.2033).

Scenario Problem A: Exceeding 20 Gallons

  • Question: What is the probability that demand exceeds 20 gallons?

    • Calculate Z: (Z = \frac{(20 - 15)}{6} = 0.8333) → round to 0.83.

    • Table lookup: Area to the left = 0.7967.

    • Probability: (0.2033) (about 20%).

Scenario Problem B: No More Than 11 Gallons

  • Question: What is the probability that demand is no more than 11 gallons?

    • Less than or equal interpretation ((X \leq 11)).

    • Calculate Z: (Z = \frac{(11 - 15)}{6} = -0.6667 \rightarrow -0.67).

    • Table lookup: Area = 0.2514.

    • Probability: (P(X \leq 11) = 0.2514).

Scenario Problem C: Demand Between 1 and 25 Gallons

  • Question: What is the probability that demand is between 1 gallon and 25 gallons?

    • Define the range: (1 < X < 25).

    • Calculate Z for 1:

      • (Z_1 = \frac{(1 - 15)}{6} = -2.3333 \rightarrow -2.33).

    • Calculate Z for 25:

      • (Z_2 = \frac{(25 - 15)}{6} = 1.6667 \rightarrow 1.67).

    • Table lookup values:

      • For (Z_1 = -2.33) → area = 0.0099.

      • For (Z_2 = 1.67) → area = 0.9525.

    • Determine the Area: (P(1 < X < 25) = 0.9525 - 0.0099 = 0.9426).

Key Takeaways

  • Always convert to Z scores for normal distribution questions.

  • Use the correct visual and shade areas based on the question type.

  • Use the table correctly to find probabilities for less than, greater than, or between scenarios.