NY Regents Algebra II: a complete guide to statistics and probability on the exam

What this strand demands

The statistics-and-probability strand is a substantial share of NY Regents Algebra II, worth roughly a fifth of the credits. It rewards working with the normal distribution and z-scores, applying the probability rules and conditional probability, reasoning clearly about study design and sampling, and fitting and diagnosing regression models. This guide ties together the dot-point pages, each with its own practice: the normal distribution and z-scores, conditional probability and the probability rules, sampling and study design, and regression and statistical inference.

The normal distribution and z-scores

A normal distribution is the symmetric bell curve, centered at its mean with spread set by the standard deviation. The empirical rule gives about 68%, 95%, and 99.7% within one, two, and three standard deviations. A z-score $z = \frac{x - \mu}{\sigma}$ measures distance from the mean in standard deviations, and converting to a z-score lets a calculator find any proportion.

Conditional probability and the rules

Conditional probability $P(A \mid B) = \frac{P(A \text{ and } B)}{P(B)}$ conditions on group B (its total is the denominator). The addition rule for "A or B" subtracts the overlap; the multiplication rule for "A and B" chains $P(A)$ with $P(B \mid A)$. Two events are independent when $P(A \mid B) = P(A)$, equivalently $P(A \text{ and } B) = P(A) \cdot P(B)$.

Sampling and study design

A survey estimates a population value from a sample; an observational study watches existing groups; an experiment imposes a treatment. Only a randomized experiment can establish causation. Random sampling reduces bias. A simulation builds a sampling distribution; its spread gives a margin of error (about two standard deviations), and the plausible range is the statistic plus or minus the margin.

Regression and inference

Regression fits linear, exponential, or other models, with parameters interpreted in context. The correlation coefficient $r$ (from $-1$ to $1$) gives the strength and direction of a linear relationship, but never causation. A residual plot diagnoses the model type: random scatter supports it, a clear pattern rejects it.

How this strand is examined

• Part I (2 credits). An empirical-rule percentage, a z-score, an addition-rule probability, a study-design classification, or an $r$ interpretation.
• Part II (2 credits). A z-score with interpretation, a conditional probability from a table, or a margin-of-error estimate. Show the setup.
• Part III and IV (4 to 6 credits). A normal-distribution proportion problem, an independence analysis, a simulation with a margin of error, or a regression with a residual-plot explanation. Show every step and interpret.

Check your knowledge

Work these as you would for credit.

1. Data is normal with mean 60, standard deviation 10. About what percent lies between 50 and 70? (1 credit)
2. Find the z-score of $x = 95$ when $\mu = 80$ and $\sigma = 5$. (2 credits)
3. $P(A) = 0.6$, $P(B) = 0.3$, $P(A \text{ and } B) = 0.2$. Find $P(A \text{ or } B)$. (2 credits)
4. Of 50 people, 30 like coffee and 12 of those also like tea. Find $P(\text{tea} \mid \text{coffee})$. (2 credits)
5. A researcher randomly assigns subjects to a treatment or placebo. What study type is this? (1 credit)
6. A simulation gives sample proportions with standard deviation 0.02. Estimate the margin of error. (2 credits)
7. Interpret $r = -0.91$. (2 credits)
8. An exponential model $\hat{y} = 100(0.9)^x$ models decay. Predict $y$ at $x = 5$, to the nearest whole number, and state whether a curved residual plot would support a linear model. (4 credits)