Summary of "WST01/01, (IAL), Edexcel, S1, Jan 2020, Q5 , Normal Distribution"

Main Ideas and Concepts

Normal Distribution Basics
- The random variable x follows a Normal Distribution with a mean (μ) of 10 and a standard deviation (σ) of 6.
- To find probabilities related to x, it is necessary to standardize the score using the formula:
  z = (x - μ) / σ
Part 5a: Probability Calculation
- The task is to find P(x < 7).
- Standardizing the score:
  z = (7 - 10) / 6 = -0.5
- Using the Normal Distribution table, the area to the left of z = 0.5 is found (0.6915), and thus:
  P(x < 7) = 1 - P(z < 0.5) = 1 - 0.6915 = 0.3085 ≈ 0.309
Part 5b: Finding k
- The problem states P(10 - k < x < 10 + k) = 0.6.
- By symmetry, the probability on either side of the mean (10) is 0.3.
- Using the Normal Distribution table, find z such that P(z > y) = 0.2, leading to z = 0.8416.
- Standardizing gives:
  y = 6 × 0.8416 + 10 ≈ 15.0496 ⇒ k ≈ 5.05
Part 5c: Area of Rectangle
- The area of the rectangle formed by coordinates is given by A = x(x - 3).
- The goal is to find P(A > 40) or P(x² - 3x - 40 > 0).
- The critical points are found by solving x² - 3x - 40 = 0, yielding x = 8 and x = -5.
- The probabilities are calculated for x < -5 and x > 8 using the z-scores:
  - For x < -5: z = (-5 - 10) / 6 ≈ -2.5 (area found as 0.0062).
  - For x > 8: z = (8 - 10) / 6 ≈ -0.33 (area found as 0.6293).
- The total probability P(A > 40) is:
  P(A > 40) ≈ 0.0062 + 0.6293 = 0.6355 ≈ 0.636

Standardization: Use the formula z = (x - μ) / σ to convert x values into z-scores.
Use of Normal Distribution Table: Identify areas corresponding to z-scores to find probabilities.
Symmetry in Normal Distribution: Utilize symmetry to simplify calculations for probabilities around the mean.
Quadratic Inequality: Solve quadratic inequalities to find critical points and determine probabilities.