Stat 226 - Lecture 9

02/11/14

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:
    • \( P(Y > -9) \)
    • \( P(-9 < Y < -7) \)

Percentiles

  • Definition: The \( p^{th} \) percentile is a value, say \( v \), where \( p \) percent of observations are lower than \( v \).

Finding Percentiles of Z

  • Remember that for the normal distribution, mean = median = mode!
  • Thus, the median of the standard normal is…
    0!
  • Therefore the 50th percentile of the standard normal is …
    0!
  • If we denote \( Z \sim N(0, 1) \), we just said that \( P(Z < 0) = 1/2 \).
  • Suppose I want to find the \( 99^{th} \) percentile of \( Z \)…
  • We have to find a value \( z^* \) that yields \( P(Z < z^*) = 0.99 \)

Your Turn

  1. Find the value of \( z^* \) for each of the following scenarios:
    • \( P(Z < z^*) = 0.025 \) (Note: \( z^* \) is called the 2.5th percentile)
    • \( P(Z < z^*) = 0.975 \) (Note: \( z^* \) is called the 97.5th percentile)
    • \( P(Z < z^*) = 0.25 \) (Note: \( z^* \) is called the 25th percentile)
  2. Find the two values that bound the middle 90% of the standard normal distribution.

Finding percentiles for any X

  • As before, \( X \sim N(\mu, \sigma) \) translates to \( Z \sim N(0,1) \) via a z-score:
\[ z = \frac{x-\mu}{\sigma} \]
  • Rearranging this equation a bit gives us:
\[ z \times \sigma + \mu = x \]
  • Thus, if \( z^* \) is the \( p^{th} \) percentile of the standard normal, then \( \mu + \sigma \times z^* \) is the \( p^{th} \) percentile of \( X \sim N(\mu, \sigma) \)!

Average Debt

  • Example: Let \( X \) be the debt of an ISU undergraduate upon graduation (in thousands of dollars). Suppose \( X \) is known to have a mean of 20 and standard deviation of 5.
    • How much debt does it take to have more debt than 95% of other undergraduates?
    • In other words, what value of \( z^* \) gives us \( P(Z < z^*) = 0.95 \)?
    • From Table A, \( z^* = \) 1.64 or 1.65 (either one is fine).
    • Using our 'backwards z-score calculation': \( x = \mu + \sigma \times z^* = 20 + 5 \times 1.65 = \) 28.25
    • So, if you have $28,500 in debt upon graduating, you have more debt than 95% of graduates!

Your Turn

  1. Suppose \( X \sim N(12, 2^2) \). Find the value of \( x^* \) for each of the following scenarios:
    • \( P(X < x^*) = 0.33 \) (\( x^* \) is called the 33rd percentile)
    • \( P(X < x^*) = 0.66 \) (\( x^* \) is called the 66th percentile)
    • \( P(X < x^*) = 0.01 \) (\( x^* \) is called the 1st percentile)
  2. Suppose monthly revenue in your business market is known to vary according to a normal distribution with mean of $500 thousand and variance of $200 thousand.
    • How much revenue do you need next month in order to sell more than 98% of the entire market?
    • What monthly revenue figure would imply that 40% of the market receieved a higher revenue?