04/28/2025

Z-Scores and Probabilities

• Z-Score Formula: The formula to calculate the z-score is given by:
  z = \frac{x - \mu}{\sigma}
  where:

  • $x$: value of the observation
  • $\mu$: mean of the distribution
  • $\sigma$: standard deviation of the distribution

• Example Calculation:

  • For values $x=132$, $\mu=120$, and $\sigma=5$, the calculation would be:
    z = \frac{132 - 120}{5} = \frac{12}{5} = 2.4

• Finding Probability:

  • The probability corresponding to $z=2.4$ is found using a z-table:
  • Result: 0.9918 (implying a 99.18% probability that a value is less than 132).

Sample Size and Z-Scores

• Z-Score with Sample Size: The formula is adjusted for sample size $n$:
  z = \frac{x - \mu}{\sigma / \sqrt{n}}

• Example Calculation:

  • For $x=306$, $\mu=300$, and $n=25$, with standard deviation $\sigma=4$:
    z = \frac{306 - 300}{4 / \sqrt{25}} = \frac{6}{0.8} = 7.5

• Probability Corresponding to Z-Score:

  • The probability of a z-score greater than 7.5 is extremely small: 0.0000007 .

Conditional Probability Concepts

• Definitions:

  • Conditional Probability: This is the probability of an event occurring given that another event has already occurred.
  • Formula:
    P(A | B) = \frac{P(A \cap B)}{P(B)}

• Example Calculation:

  • For a female who answers yes, probability computed as:
    P(\text{Yes and Female}) = \frac{27}{66} \approx 0.409 .

Hypothesis Testing and Critical Values

• Critical Value Calculation:

  • Given $\alpha = 0.05$, critical value z-score would be:
    z_{critical} = 1.96
  • Usage: For hypothesis testing, check if test statistic falls in the rejection region.

• Test Setup: Null hypothesis (H0) and alternative hypothesis (H1) defined as:

  • H0: mean = 22.1
  • H1: mean < 22.1 (for a one-tailed test)

• Test Statistic Calculation:

  • t = \frac{\bar{x} - \mu}{s / \sqrt{n}}
  • Example Calculation:
    • For given values, x = 21.5, \mu=22.1, s=0.2, n=20 :
    • t \approx -13.4 , which would result in rejecting H0.

Linear Regression Calculations

• Linear Regression Equation:

  • The formula for the line of best fit is represented as:
    Y = B0 + B1 X

• Example Calculation:

  • If $B0 = 4.11$ and $B1 = 0.637$, then plugging $X=8$ yields:
    Y_{hat} = 0.637(8) + 4.11 \approx 9.214
  • Residual calculation: 4 - 9.214 = -5.214 .

Combinatorics and Binomial Probability

• Combinations Formula: To determine the number of ways to choose $x$ successes out of $n$ trials:
  C(n, x) = \frac{n!}{x!(n-x)!}

• Binomial Probability Formula:

  • The probability of exactly $x$ successes is given by:
    P(X = x) = C(n, x) p^x (1 - p)^{n - x}
  • For example, with $n=12$, $x=3$, and $p=0.66$, use binomial calculations:
  • $C(12, 3) imes (0.66)^3 imes (0.34)^{9}$.
  • Calculator steps involve the binom PDF function, returning probabilities immediately.