AP Statistics Exam Study Guide

AP Statistics Cumulative AP Exam Study Guide

Statistics Basics

• Statistics: Science of collecting, analyzing, and drawing conclusions from data.
• Descriptive Statistics: Methods for organizing and summarizing data.
• Inferential Statistics: Making generalizations from a sample to a population.
• Population: Entire collection of individuals or objects.
• Sample: Subset of the population selected for study.
• Variable: Characteristic whose value changes.
• Data: Observations on single or multi-variables.

Types of Variables

• Categorical (Qualitative): Basic characteristics.
• Numerical (Quantitative): Measurements or observations of numerical data.
  • Discrete: Listable sets (counts).
  • Continuous: Any value over an interval (measurements).
• Univariate: One variable.
• Bivariate: Two variables.
• Multivariate: Many variables.

Distributions

• Symmetrical: Data with fairly same shape and size on both sides.
• Uniform: Every class has equal frequency.
• Skewed: One side (tail) is longer than the other. Skewness is in the direction the tail points.
• Bimodal: Two or more classes have large frequencies separated by another class.

Describing Numerical Graphs (S.O.C.S.)

• Shape: Symmetrical, skewed (right/left), uniform, or bimodal.
• Outliers: Gaps, clusters, etc.
• Center: Middle of the data (mean, median, mode).
• Spread: Variability (range, standard deviation, IQR).
• Context: Everything must be in context to the data and situation.
• Comparison: When comparing distributions, use comparative language.

Parameters vs. Statistics

• Parameter: Value of a population (typically unknown).
• Statistic: Calculated value about a population from a sample(s).

Measures of Center

• Median: Middle point of the data (50th percentile) in numerical order.
• Mean: $μ$ for population, $\bar{x}$ for sample.
• Mode: Occurs most in the data. Can have multiple modes or none.

Measures of Spread (Variability)

• Range: $Max - Min$.
• IQR: Interquartile range $(Q3 - Q1)$.
• Standard Deviation: $σ$ for population, $s$ for sample. Measures typical deviation from the mean. Sample standard deviation is divided by $df = n-1$.
• Sum of deviations from the mean is always zero.
• Variance: Standard deviation squared.

Resistant vs. Non-Resistant Measures

• Resistant: Not affected by outliers (Median, IQR).
• Non-Resistant: Affected by outliers (Mean, Range, Variance, Standard Deviation, Correlation Coefficient (r), Least Squares Regression Line (LSRL), Coefficient of Determination $(r^2)$).

Comparison of Mean & Median Based on Graph Type

• Symmetrical: Mean and median are the same.
• Skewed Right: Mean > Median.
• Skewed Left: Mean < Median.
• Mean is pulled in the direction of the skew.
• Trimmed Mean: Use a % to remove observations from the top and bottom to eliminate outliers.

Linear Transformations of Random Variables

• $μ<em>{a +bx} =a +bμ</em>x$ (Mean is changed by both addition/subtraction & multiplication/division).
• $σ<em>{a +bx} = |b|σ</em>x$ (Standard deviation is changed by multiplication/division ONLY).

Combination of Two (or More) Random Variables

• $μ<em>{x ± y} = μ</em>x ± μ_y$ (Add or subtract the means).
• $σ^2<em>{x ± y} = σ^2</em>x + σ^2_y$ (Always add the variances - X & Y MUST be independent).

Z-Score

• Standardized score indicating how many standard deviations an observation is from the mean. Creates a standard normal curve with $μ = 0$ & $σ = 1$.
• $z = {x - μ \over σ}$

Normal Curve

• Bell-shaped and symmetrical.
• As $σ$ increases, the curve flattens.
• As $σ$ decreases, the curve thins.

Empirical Rule (68-95-99.7)

• Measures $1σ$, $2σ$, and $3σ$ on normal curves from the center $μ$.
• 68% of the population is between $-1σ$ and $1σ$.
• 95% of the population is between $-2σ$ and $2σ$.
• 99.7% of the population is between $-3σ$ and $3σ$.

Boxplots

• For medium or large numerical data; doesn't contain original observations.
• Use modified boxplots with fences at $1.5 * IQR$ from the ends of the box (Q1 & Q3).
• Points outside the fence are outliers.
• Whiskers extend to the smallest & largest observations within the fences.

5-Number Summary

• Minimum, Q1 (25th Percentile), Median, Q3 (75th Percentile), Maximum

Probability Rules

• Sample Space: Collection of all outcomes.
• Event: Any sample of outcomes.
• Complement: All outcomes not in the event.
• Union: A or B, all outcomes in both circles. $A ∪ B$
• Intersection: A and B, happening in the middle of A and B. $A ∩ B$
• Mutually Exclusive (Disjoint): A and B have no intersection; they cannot happen at the same time.
• Independent: Knowing one event doesn't change the outcome of another.
• Experimental Probability: Number of successes from an experiment divided by the total amount from the experiment.
• Law of Large Numbers: As an experiment is repeated, the experimental probability gets closer to the true probability.

Probability Rules (Formulas)

• All values are 0 < P < 1.
• Probability of sample space is 1.
• Complement: $P + (1 - P) = 1$
• Addition: P(A or B) = P(A) + P(B) – P(A & B)
• Multiplication: P(A & B) = P(A) * P(B) if A & B are independent.
• $P (at least 1 or more) = 1 – P (none)$
• Conditional Probability: P(A|B) = {P(A & B) \over P(B)}

Correlation Coefficient

• (r) - Quantitative assessment of the strength and direction of a linear relationship. (use $ρ$ (rho) for population parameter)
• Values: $[-1, 1]$ ; 0 - no correlation, $(0, ±0.5)$ - weak, $[±0.5, ±0.8)$ - moderate, $[±0.8, ±1]$ - strong

Least Squares Regression Line (LSRL)

• Line of best fit. Minimizes deviations (residuals) from the line. Used with bivariate data.
• $\hat{y} = a + bx$; x is independent, y is dependent.
• Residuals (error) - vertical difference of a point from the LSRL. All residuals sum to 0.
• Residual = $y - \hat{y}$
• Residual Plot - scatterplot of $(x, residual)$. No pattern indicates a linear relationship.

Coefficient of Determination

• $(r^2)$ - Proportion of variation in y explained by the relationship of (x, y). Never use the adjusted $r^2$.
• Interpretations: (must be in context!)
  • Slope (b): For unit increase in x, the y variable will increase/decrease by the slope amount.
  • Correlation coefficient (r): There is a (strength, direction, linear) association between x & y.
  • **Coefficient of determination $(r^2)$: Approximately $r^2$% of the variation in y can be explained by the LSRL of x and y.
• Extrapolation - LRSL cannot be used to find values outside the range of the original data.
• Influential Points - if removed, significantly change the LSRL.
• Outliers - points with large residuals.

Census

• A complete count of the population. Why not to use a census?
  • Expensive
  • Impossible to do
  • If destructive sampling you get extinction

Sampling Frame

• Is a list of everyone in the population.

Sampling Design

• Refers to the method used to choose a sample.

SRS (Simple Random Sample)

• One chooses so that each unit has an equal chance and every set of units has an equal chance of being selected.
  • Advantages: Easy and unbiased
  • Disadvantages: Large $σ^2$ and must know population.

Stratified

• Divide the population into homogeneous groups called strata, then SRS each strata.
  • Advantages: More precise than an SRS and cost reduced if strata are already available.
  • Disadvantages: Difficult to divide into groups, more complex formulas & must know population.

Systematic

• Use a systematic approach (every 50th) after choosing randomly where to begin.
  • Advantages: Unbiased, the sample is evenly distributed across the population & don’t need to know population.
  • Disadvantages: A large $σ^2$ and can be confounded by trends.

Cluster Sample

• Based on location. Select a random location and sample ALL at that location.
  • Advantages: Cost is reduced, unbiased & don’t need to know the population.
  • Disadvantages: May not be representative of the population and has complex formulas.

Random Digit Table

• Each entry is equally likely, and each digit is independent of the rest.

Random # Generator

• Calculator or computer program

Bias

• Error that favors a certain outcome related to the center of sampling distributions.

Sources of Bias

• Voluntary Response: People choose themselves to participate.
• Convenience Sampling: Ask people who are easy or comfortable to ask.
• Undercoverage: Some group(s) are left out of the selection process.
• Non-response: Someone cannot or does not want to participate.
• Response: False answers due to question wording.

Experimental Design

• Observational Study: Observe outcomes without giving a treatment.
• Experiment: Actively impose a treatment on the subjects.
• Experimental Unit: Single individual or object that receives a treatment.
• Factor: Explanatory variable being tested.
• Level: A specific value for the factor.
• Response Variable: What you are measuring with the experiment.
• Treatment: Experimental condition applied to each unit.
• Control Group: Group used to compare the factor to for effectiveness (doesn't have to be placebo).
• Placebo: Treatment with no active ingredients.
• Blinding: Subjects are unaware of the treatment.
• Double Blinding: Neither subjects nor evaluators know which treatment is being given.

Principles of Experimental Design

• Control: Keep all extraneous variables constant.
• Replication: Use many subjects to quantify the natural variation in the response.
• Randomization: Use chance to assign subjects to treatments.
• The only way to show cause and effect is with a well designed, well controlled experiment.

Experimental Designs

• Completely Randomized: All units are randomly allocated to all treatments.
• Randomized Block: Units are blocked and then randomly assigned within each block (reduces variation).
• Matched Pairs: Units are matched and then randomly assigned. OR individuals do both treatments in random order (assignment is dependent).
• Confounding Variables: Effect of the variable on the response cannot be separated from the factor being tested - happens in observational studies.
• Randomization reduces bias by spreading extraneous variables to all groups.
• Blocking helps reduce variability. Another way to reduce variability is to increase sample size.

Random Variable

• A numerical value that depends on the outcome of an experiment.

Discrete Probability Distributions

• Gives values & probabilities associated with each possible x.
• $μ<em>X = Σx</em>i * p(x_i)$
• $σ^2<em>X = Σ(x</em>i - μ)^2 * p(x_i)$
• Fair Game = All pay-ins equal all pay-outs.

Special Discrete Distributions

• Binomial Distributions:

  • Two mutually exclusive outcomes, fixed number of trials (n), each trial is independent, probability (p) of success is the same for all trials.
  • Random variable - is the number of successes out of a fixed # of trials. Starts at X = 0 and is finite.
  • $μ_X = np$
  • $σ_x = \sqrt{npq}$
  • Calculator: binomialpdf (n, p, x) = single outcome P(X= x), binomialcdf (n, p, x) = cumulative outcome P(X < x), 1 - binomialcdf (n, p, (x -1)) = cumulative outcome P(X > x)

• Geometric Distributions:

  • Two mutually exclusive outcomes, each trial is independent, probability (p) of success is the same for all trials. (NOT a fixed number of trials)
  • Random Variable –when the FIRST success occurs. Starts at 1 and is ∞.
  • Calculator: geometricpdf (p, a) = single outcome P(X = a), geometriccdf (p, a) = cumulative outcomes P(X < a), 1 - geometriccdf (n, p, (a -1)) = cumulative outcome P(X > a)

Continuous Random Variable

• Numerical values that fall within a range or interval (measurements). Area under the curve always = 1. To find probabilities, find the area under the curve.

Unusual Density Curves

• Any shape (triangles, etc.)

Uniform Distributions

• Uniformly (evenly) distributed, shape of a rectangle

Normal Distributions

• Symmetrical, unimodal, bell shaped curves defined by the parameters $μ$ & $σ$.
• Calculator:
  • Normalpdf – used for graphing only
  • Normalcdf(lower bound, upper bound, μ, σ) – finds probability
  • InvNorm(p) – z-score OR InvNorm(p, μ, σ) – gives x-value
• To assess Normality - Use graphs – dotplots, boxplots, histograms, or normal probability plot.
• Distribution – is all of the values of a random variable.
• Sampling Distribution – of a statistic is the distribution of all possible values of all possible samples. Use normalcdf to calculate probabilities.

Sampling Distributions

• $\mu<em>{\bar{X}} = μ</em>X$
• $σ_{\bar{x}} = {σ \over \sqrt{n}}$ (standard deviation of the sample means)
• $\mu_{\hat{p}} = p$
• $σ_{\hat{p}} = \sqrt{{pq \over n}}$ (standard deviation of the sample proportions)
• $\mu<em>{X</em>1 - X<em>2} = μ</em>{X<em>1} - μ</em>{X_2}$
• $σ<em>{X</em>1 - X<em>2} = \sqrt{{σ</em>1^2 \over n<em>1} + {σ</em>2^2 \over n_2}}$ (standard deviation of the difference in sample means)
• $\mu<em>{\hat{p</em>1} - \hat{p<em>2}} = p</em>1 - p_2$
• $σ<em>{\hat{p</em>1} - \hat{p<em>2}} = \sqrt{{{p</em>1q<em>1} \over n</em>1} + {{p<em>2q</em>2} \over n_2}}$ (standard deviation of the difference in sample proportions)
• $\mu_b = β$
• $s_b$ (do not need to find, usually given in computer printout (standard error of the slopes of the LSRLs)
• Standard error – estimate of the standard deviation of the statistic

Central Limit Theorem

• When n is sufficiently large (n > 30) the sampling distribution is approximately normal even if the population distribution is not normal.

Confidence Intervals

• Point Estimate: Uses a single statistic based on sample data.
• Confidence Intervals: Used to estimate the unknown population parameter.
• Margin of Error: The smaller the margin of error, the more precise our estimate.
• Steps:
  • Assumptions – see table below
  • Calculations – C.I. = statistic ± critical value * (standard deviation of the statistic)
  • Conclusion – Write your statement in context. We are [x]% confident that the true [parameter] of [context] is between [a] and [b].
• What makes the margin of error smaller:
  • Make critical value smaller (lower confidence level).
  • Get a sample with a smaller s.
  • Make n larger.

T distributions compared to standard normal curve

• Centered around 0
• More spread out and shorter
• More area under the tails.
• When you increase n, t-curves become more normal.
• Can be no outliers in the sample data
• Degrees of Freedom = n – 1
• Robust – if the assumption of normality is not met, the confidence level or p-value does not change much – this is true of t-distributions because there is more area in the tails

Hypothesis Tests

• Tells us if a value occurs by random chance or not. If it is unlikely to occur by random chance then it is statistically significant.
• Null Hypothesis: $H_0$ is the statement being tested. Null hypothesis should be “no effect”, “no difference”, or “no relationship”
• Alternate Hypothesis: $H_a$ is the statement suspected of being true.
• P-Value: Assuming the null is true, the probability of obtaining the observed result or more extreme
• Level of Significance: $α$ is the amount of evidence necessary before rejecting the null hypothesis.
• Steps:
  • Assumptions – see table below
  • Hypotheses - don’t forget to define parameter
  • Calculations – find z or t test statistic & p-value
  • Conclusion – Write your statement in context. Since the p-value is < (>) α, I reject (fail to reject) the Ho. There is (is not) sufficient evidence to suggest that [Ha].

Type I and II Errors and Power

• Type I Error: Reject $H<em>0$ when $H</em>0$ is actually true (probability is $α$).
• Type II Error: Fail to reject $H<em>0$, and $H</em>0$ is actually false (probability is $β$).
• $α$ and $β$ are inversely related. Consequences are the results of making a Type I or Type II error.
• The Power of a Test – is the probability that the test will reject the null hypothesis when the null hypothesis is false assuming the null is true. Power = 1 – $β$

$\chi^2$ Test

• Used to test counts of categorical data.
  • Goodness of Fit (univariate)
  • Independence (bivariate)
  • Homogeneity (univariate 2 (or more) samples)
• $\chi^2$ Distribution: All curves are skewed right, every df has a different curve, and as the degrees of freedom increase the $\chi^2$ curve becomes more normal.
• Goodness of Fit: Univariate categorical data from a single sample. Does the observed count “fit” what we expect? Must use list to perform, df = number of the categories – 1, use χ2cdf (χ2, ∞, df) to calculate p-value
• Independence: Bivariate categorical data from one sample. Are the two variables independent or dependent? Use matrices to calculate
• Homogeneity: Single categorical variable from 2 (or more) samples. Are distributions homogeneous? Use matrices to calculate

For both $\chi^2$ tests of independence & homogeneity:

• Expected counts = ${(row total)(column total) \over grand total}$
• df = (r – 1)(c – 1)

Regression Model

• X & Y have a linear relationship where the true LSRL is $μ_y = α + βx$
• The responses (y) are normally distributed for a given x-value.
• The standard deviation of the responses ($\sigma<em>y$) is the same for all values of x. o S is the estimate for $\sigma</em>y$

Confidence Interval

• $b ± t^*s_b$

Hypothesis Testing

• ${b - β \over s_b}$

Assumptions: