CourseKata Chapter 6: Part III (Sec 5 to 11)

Sampling

Mansour Abdoli, PhD

Today: Normal Approximation & Applications

Session Goals

By the end of today, you can:

  • Find a Normal Approximation of an Empty-Model
  • Evaluate the Normal Approximation
  • Use the Normal Approximation to Estimate Event Propbabilities
  • Compute Z-Score
  • Use Z-Score and Probability to Compare Hypotheses

Empirical Rule is an Approximation

Normal N(mu, sigma) with vertical lines at 1, 2, and 3 sigma away from mu
- Area within 1 SD from the mean:  0.6826895 
- Area within 2 SD from the mean:  0.9544997 
- Area within 3 SD from the mean:  0.9973002

Fitting a Normal to a Distribution

Student Thumb Length:

Evaluate the Fit

Normal Approximation of Student Thumb Length:

  • Recall Empirical Rule
    • \(P(\mu-\sigma, \mu+\sigma)=68\%\)
    • \(P(\mu-2\sigma, \mu+2\sigma)=95\%\)
    • \(P(\mu-3\sigma, \mu+3\sigma)=99.7\%\)

Thumb-length histogram super imposed with a normal curve.

  • Proportions:
1-SD: 0.6942675      2-SD: 0.9363057     3-SD: 0.9872611
  • Not a Great Approximation; but OK!

Normal Distribution

Generic vs Standard Normal

  • Recall Empirical Rule
    • \(P(\mu-\sigma, \mu+\sigma)=68\%\)
    • \(P(\mu-2\sigma, \mu+2\sigma)=95\%\)
    • \(P(\mu-3\sigma, \mu+3\sigma)=99.7\%\)

  • Each Normal is equivalent to Standard Normal

\[X\sim N(\mu, \sigma) \Leftrightarrow Z\sim N(0, 1)\]

\[X = \mu + Z\cdot \sigma \Leftrightarrow Z=\frac{X-\mu}{\sigma}\]

Standardization

Interpretation of \(Z\)

  • \(Z=\frac{X-\mu}{\sigma}\):   \(X\) is \(Z\)-many standard deviation above \(\mu\)

  • Example: \(T\sim N(75, 5)\):

    • \(T=82\)
    • Equivalant \(Z=1.4 (=\frac{82-75}{5})\)
    • \(T=82\) is \(1.4\) standard deviation above the mean

Application of \(Z\):
Within Group Comparison

  Gender     mean       sd
1 female 64.51830 2.662054
2   male 69.50222 2.937647
  • Which one is taller in their group:
    • A 68-inch female student
    • A 72-inch male student

Female: Z = (68.00-64.52)/(2.662) 
    = 1.308

Male: Z = (72.00-69.50)/(2.938) 
    = 0.850

Application of \(Z\):
Between Group Decision

  • A student is 68 inches; is the student male or female?

  • Which is more unlikely?
    • From female group? \(P(Height > 68)\)
    • From male group? \(P(Height < 68)\)
Female: P(Height>68)=P(Z>1.31)=0.0893
Male: P(Height<68)=P(Z<-0.51)=0.2667