04/28/2025

Z-Scores and Probabilities

  • Z-Score Formula: The formula to calculate the z-score is given by:
    z=x−μσz = \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=132−1205=125=2.4z = \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=x−μσ/nz = \frac{x - \mu}{\sigma / \sqrt{n}}

  • Example Calculation:

    • For $x=306$, $\mu=300$, and $n=25$, with standard deviation $\sigma=4$:
      z=306−3004/25=60.8=7.5z = \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: 0.00000070.0000007.

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)=P(A∩B)P(B)P(A | B) = \frac{P(A \cap B)}{P(B)}

  • Example Calculation:

    • For a female who answers yes, probability computed as:
      P(Yes and Female)=2766≈0.409P(\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:
      zcritical=1.96z_{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=xˉ−μs/nt = \frac{\bar{x} - \mu}{s / \sqrt{n}}
    • Example Calculation:
      • For given values, x=21.5,μ=22.1,s=0.2,n=20x = 21.5, \mu=22.1, s=0.2, n=20:
      • t≈−13.4t \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=B<em>0+B</em>1XY = B<em>0 + B</em>1 X

  • Example Calculation:

    • If $B0 = 4.11$ and $B1 = 0.637$, then plugging $X=8$ yields:
      Yhat=0.637(8)+4.11≈9.214Y_{hat} = 0.637(8) + 4.11 \approx 9.214
    • Residual calculation: 4−9.214=−5.2144 - 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)=n!x!(n−x)!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)px(1−p)n−xP(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.