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
- For values $x=132$, $\mu=120$, and $\sigma=5$, the calculation would be:
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
- For $x=306$, $\mu=300$, and $n=25$, with standard deviation $\sigma=4$:
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 .
- For a female who answers yes, probability computed as:
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.
- Given $\alpha = 0.05$, critical value z-score would be:
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
- The formula for the line of best fit is represented as:
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 .
- If $B0 = 4.11$ and $B1 = 0.637$, then plugging $X=8$ yields:
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.
- The probability of exactly $x$ successes is given by: