Stat 226 - Lecture 8

02/06/14

Back to Empirical Rule

  • OK, but what if I want to find something like \( P(Z < -0.5) \)?
  • This is what Table A is for!

Visualizing Table A

Some useful facts

  • Using Table A requires reducing probability statements down to \( P(Z < a) \), for some number \( a \).
  • Given \( X \sim N(\mu, \sigma^2) \):
    • \( P(X > a) = 1 - P(X < a) \)
    • \( P(a < X < b) = P(X < b) - P(X < a) \)
  • Given \( Z \sim N(0, 1) \) for any positive number \( a \):
    • \( P(Z > a) = P(Z < -a) \)

Your turn

  • Find the following probabilities using Table A:
    • \( P(Z < -1.64) \)
    • \( P(Z > -0.56) \)
    • \( P(-1 < Z < 2.27) \)
    • \( P(Z > 4) \)

Finding probabilities for any X

  • Last time we learned how \( X \sim N(\mu, \sigma) \) translates to \( Z \sim N(0,1) \) via a z-score:
\[ z = \frac{x-\mu}{\sigma} \]

  • Suppose \( X \sim N(10, 25) \).

  • \( P(X > 15) = \)

    \( P(X-10 > 5) = \)
    \( P(\frac{X-10}{5} > \frac{15-10}{5}) = \)
    \( P(Z > 1) \)

  • \( P(Z > 1) = \)
    \( P(Z < -1) = \)
    .1587

Your Turn

  • Find the following probabilities using Table A:
  • Given \( X \sim N(150, 100) \), find:
    • \( P(X < 137) \)
    • \( P(X > 137) \)
    • \( P(140 < X < 165) \)
  • Given \( Y \sim N(-10, 3^2) \), find:
    • \( P(Y > -9) \)
    • \( P(-9 < Y < -7) \)