pt 2: Normal Distribution and Percentiles

Chapter 6: Normal Distribution - Review and Examples

• Review of Formulas:

  • The formula sheet contains expected value formulas for binomial distributions and formulas from Chapter 6.
  • Key formula: Transformation formula: $z = \frac{x - \mu}{\sigma}$, where:
    • $x$ = observation
    • $\mu$ = mean
    • $\sigma$ = standard deviation

• Normal Distribution Table:

  • Use either the table or calculator for z-values.
  • Calculators have a built-in z-function.

• Probability Calculations:

  • Problems involve finding P(X < value).
  • Given: X has a normal distribution, with $\sigma$ and $\mu$.

• Transformation:

  • Apply the transformation formula: $z = \frac{x - \mu}{\sigma}$.
  • Convert X to Z, then use the calculator.

• Area Representation:

  • Plot the mean in the middle.
  • The given value (e.g., 18.6) is located relative to the mean.
  • The inequality sign indicates the area of interest.

• Complement Rule:

  • If calculating probabilities to the right, use the complement rule: 1 - P(X < value).
  • Transform to Z, then calculate.

• Conditional Probability Example:

  • Given download time is more than 17.3 seconds, find the probability it's less than 19.2 seconds.

  • Apply the conditional probability rule: $P(A|B) = \frac{P(A \cap B)}{P(B)}$

  • Where:

    • $A$: X < 19.2
    • $B$: X > 17.3

  • Overlapping Events:

    • Determine the overlap of events.
    • If no overlap, events are mutually exclusive.
    • If overlap exists, calculate the intersection.

  • Calculation:

    • P(17.3 < X < 19.2) = P(X < 19.2) - P(X < 17.3)
    • Transform X values to Z values using $z = \frac{x - \mu}{\sigma}$.
    • Use the calculator to find probabilities.

• Calculator Usage:

  • Input Z-values into the calculator to find probabilities.
  • Example: For z = 0.24, probability = 0.5948.

• Table Usage:

  • The table contains negative and positive values.
  • Find the Z-value in the table to read off the probability.

• Marginal and Joint Probabilities:

  • Reference to marginal and joint probabilities is made.

• Important Note:

  • Calculating probabilities where $X = value$ results in zero because there is no area.

• Expected Values and Variances:

  • For normal distribution, mean and standard deviation are typically given.

Importance and Applications

*P-values
* These concepts are important for p-values, hypothesis tests, and confidence intervals in later courses.

*Regression
* Normal distribution underlies regression models.

Percentiles

• Percentiles are the inverse of probability calculations.

• Formula:

  • To calculate percentiles, use: $x = \mu + z_{\alpha} \cdot \sigma$
  • Where $\alpha$ represents a given probability.

• Alpha:

  • $\alpha = 0.5$ corresponds to the median.
  • $\alpha = 0.25$ corresponds to the first quartile.
  • $\alpha = 0.75$ corresponds to the third quartile.

• Percentile Calculation:

  • Given a probability, find the corresponding X value.

Examples of Percentile Calculations

• Definitions:

  • Percentile: General term.
  • Quartiles: Specific percentiles (25th, 50th, 75th).
  • $P = 80$ (80th percentile), $\alpha = 0.8$

• Example 1: Finding the 80th Percentile of Download Time.

  • Given: X = download time, normally distributed with $\mu = 18$ and $\sigma = 5$.

  • The 80th percentile divides the distribution such that 80% is to the left.

  • Calculation:

    • $P<em>{80} = \mu + z</em>{80} \cdot \sigma$
    • Find $z_{80}$ using a calculator (inverse normal function).
    • $z_{80} \approx 0.8416$
    • $P_{80} = 18 + (0.8416)(5) = 22.208$

  • Interpretation: 80% of downloads are less than 22.208 seconds.

• Example 2: Finding X Value for a Given Probability.

  • Find X such that 5% of download times are less than X.

  • Percentile: $P_5$. $\alpha = 0.05$

  • Calculation:

    • $x = \mu + z_{0.05} \cdot \sigma$
    • Find $z_{0.05}$ using a calculator (inverse normal function).
    • $z_{0.05} \approx -1.6449$
    • $x = 18 + (-1.6449)(5) = 9.7755$

  • 5% of download times will be less than 9.7755 seconds.

• Example 3: Quartiles.

  • Quartiles divide data into four parts.

  • Calculations:

    • $Q<em>1$ (25th percentile): $x = \mu + z</em>{0.25} \cdot \sigma$ , The value for $z_{0.25} \approx -0.6745$, $x = 18 + (-0.6745)(5) = 14.6275$
    • $Q_2$ (50th percentile/median): It's equal to the mean which is 18.
    • $Q<em>3$ (75th percentile): $x = \mu + z</em>{0.75} \cdot \sigma$, The value for $z_{0.75} \approx 0.6745$, $x = 18 + (0.6745)(5) = 21.3725$

Interquartile range
* Calculate the interquartile range (IQR): $Q<em>3 - Q</em>1 = 21.3725 - 14.6275 = 6.745$
* It means that 50% of the download time will be between 14.63 and 21.37 seconds.

• Example 4: Symmetry and Probability.

  • Find A and B such that probability between A and B is 90%, symmetrically around the mean.

  • Setup:

    • P(A < X < B) = 0.90, is set symmetrically around $\mu$.

    • The area to the left of A is 0.05, and to the right of B is 0.05.

    • A is the 5th percentile, and B is the 95th percentile.

  • Formula

    • $A = \mu + z<em>{0.05} \cdot \sigma, -1.64$ , and $B = \mu + z</em>{0.95} \cdot \sigma, 1.64$