AP Statistics Cram Chart Notes

Exploring One-Variable Data

• Categorical data: two-way tables & bar graphs, analyzing proportions

• Quantitative data: histograms, dotplots, box plots, stem and leaf plots, and scatterplots.

• Mean: non-resistant, Median: resistant

• Unimodal, Bimodal, Uniform

• Compare distributions using comparison words

• Histogram: approximate the mean and use words like “no more” / “approximately” when describing range

• Analyze distributions using CUSS in context: Center, Unusual features, Shape, Spread (skew pulls mean)

• Normal distribution: mound-shaped and symmetric $(\mu)$, $(\sigma)$

• Z-score: $\frac{value - mean}{SD}$

• Standard Normal Distribution: Mean of 0, SD of 1

• Empirical Rule: 68% (1 SD), 95% (2 SD), 99.7% (3 SD)

• Categorical data independence: conditional frequency = marginal frequency

Exploring Two-Variable Data

• Describe associations with direction, strength, form.

  • Direction: positive / negative (slope)

  • Form: linear / non-linear

• r (correlation coefficient): strength & direction, NOT FORM

• Least Squares regression line (LSRL): predicts values of response variable (y) given explanatory variable (x)

• LSRL: $ŷ = a + bx$

  • ŷ = predicted value of rsp variable

  • a = y-int, b = slope

• Residual = predicted - actual

• Random scatter on residual plot

• Extrapolation: predictions outside the interval of values of x

• S & R-sq influenced by outliers (s ↑, r-sq ↓)

• Interpretations:

  • Slope/b: As the [exp var.] increases by 1 [unit], the [rsp var.] is predicted to increase by b [units].

  • Y-intercept: When there are zero [exp var], the predicted [rsp var.] is y-int.

  • s: When using LSRL to predict [rsp var] from [exp var] we are typically off by [value of s].

  • r2 (in %): About [r-sq]% of variation in [rsp var] is explained by the LSRL using [exp var].

  • Residual: The actual (rsp var) is about [residual] more/less than the predicted (rsp var).

Collecting Data

• Simple Random Sample (SRS): every group has an equal chance

• Cluster Sample: Divide pop. into heterogeneous groups [all from some]

• Stratified Random Sample: Divide pop. into strata of homogeneous groups [some from all]

• Stratifying: ↓ variability, ↑ precision

• Bias: undercoverage, nonresponse, response bias (inaccurate)

• EXPERIMENTS ASSIGN TREATMENTS

• Confounding: variable and the exp. variables affect the rsp. Variable and can’t be distinguished

• Experiments have comparison, random assignment, control, & replication

• Randomized block design: random assignment in each block

• Blocks share a var that may impact rsp, ↓ variability, allows for easier comparison

• Matched pairs: compare 2 treatments in block size 2

Probability, Random Variables, and Probability Distributions

• Probability: chance of an event occurring (0-1)

• $P(event) = \frac{successful}{total}$

• Complement: $P(not event) = 1 - P(event)$

• $P(A and B) = P(A∩B)$

• Conditional probability: $P(A) * P(B|A)$

• $P(A or B) = P(A U B) = P(A) + P(B) - P(A and B)$

• Conditional Probability: $P(A|B) = \frac{P(A and B)}{P(B)}$

• Mutually exclusive: $P(A or B) = P(A) + P(B)$

• Independent: $P(A|B) = P(A)$

• Random variables: quantitative, numerical values

• Discrete: only “whole” values possible

• Expected Value: $x1p1 + x2p2 + … + xnpn$

• Continuous: all values in an interval

• Binomial: Binary, Independent (10%), Number of fixed trials, Same probability of success p

• Geometric: Binary, Independent, same prob of success p, without trials set previously

Sampling Distributions

• Sample statistics estimate population parameters (x-bar, p-hat)

• Sampling distribution: distribution of values taken by a statistic in all possible samples

• Larger samples are less variable and more accurate

• Sampling distrib. of p-hat: approx normal if np >= 10 AND n(1-p) >= 10 (Large Counts Condition)

• Sampling distrib of x-bar: Check 10%/random condition, and n > 30

• Conditions: 10 of each, random, 10%

• Conf. Interval = Point Estimate +/- Margin of Error

• Larger sample size decreases margin of error!

• 1-sample CI: one-sample C% Z-Interval for p, hypothesis test = one sample z-test for p

• 2-sample CI: two-sample C% Z-Interval for p1 - p2, hypothesis test = two-sample z-test for p1 - p2

• NO paired data for props.

• State parameter(s) & hypotheses!

• 2-sided Ha: confidence interval to make a decision about H0. Reject if H0 not in interval [Conf level = opp of signif level)

• P-value: Assuming the [H0 in context] is true, the probability that the [observed statistic - x-bar or p-hat] will take a value as or more extreme than it does is [p-value].

• Confidence level: If we were to select many random samples and construct a C% confidence interval using each sample, about C% of the intervals would capture the [parameter in context].

• Confidence intervals: We are C% confident that the interval from __ to ___ captures the [true parameter in context]

• Conditions - n > 30, random (no experiments), independent

• Confidence interval for one sample: one-sample C% t-interval for mu, hypothesis test = one-sample t-test for μ

• Confidence interval of two samples: 2-sample C% t-interval for μ1- μ2, hyp. test = two-sample t-test for μ1- μ2

• Paired data: experimental unit received 2 treatments. mu_diff

• CI = One-sample t C% CI for μdiff, Hyp test = One-sample t test for μdiff

• Type I Error: Rejecting H0 when we should’ve failed to reject H0

• Type II error: Failing to reject the H0 when we should’ve rejected H0

• Power = 1 - P(Type II error)

• To increase power: increase sample size, increase significance level, increase the diff b/w H0 and true Ha.

Chi-Square Inference

• Formula: 3 Tests: GoF, Independence, and Homogeneity

• Hypothesis Test

• Chi-Squared is a non-parametric test

• Calculate df (n-1)

• Conditions: random, indep., at least 5 success/fail

• Name the correct type of test

Inference for Slopes

• LSRL Equation: $μ = a + βx$

• Conditions: Linear, Indep, Normal, equal SD, Random

• Confidence interval: $b ± t*(SEb)$

• df=n-2

• Hypothesis test: $(\frac{B-β0}{SEb})$

Exploring One-Variable Data

• Categorical data: two-way tables & bar graphs, analyzing proportions

• Quantitative data: histograms, dotplots, box plots, stem and leaf plots, and scatterplots.

• Mean: non-resistant, Median: resistant

• Unimodal, Bimodal, Uniform

• Compare distributions using comparison words

• Histogram: approximate the mean and use words like “no more” / “approximately” when describing range

• Analyze distributions using CUSS in context: Center, Unusual features, Shape, Spread (skew pulls mean)

• Normal distribution: mound-shaped and symmetric $(\mu)$, $(\sigma)$

• Z-score: $\frac{value - mean}{SD}$

• Standard Normal Distribution: Mean of 0, SD of 1

• Empirical Rule: 68% (1 SD), 95% (2 SD), 99.7% (3 SD)

• Categorical data independence: conditional frequency = marginal frequency

Exploring Two-Variable Data

• Describe associations with direction, strength, form.

  • Direction: positive / negative (slope)

  • Form: linear / non-linear

• r (correlation coefficient): strength & direction, NOT FORM

• Least Squares regression line (LSRL): predicts values of response variable (y) given explanatory variable (x)

• LSRL: $ŷ = a + bx$

  • ŷ = predicted value of rsp variable

  • a = y-int, b = slope

• Residual = predicted - actual

• Random scatter on residual plot

• Extrapolation: predictions outside the interval of values of x

• S & R-sq influenced by outliers (s ↑, r-sq ↓)

• Interpretations:

  • Slope/b: As the [exp var.] increases by 1 [unit], the [rsp var.] is predicted to increase by b [units].

  • Y-intercept: When there are zero [exp var], the predicted [rsp var.] is y-int.

  • s: When using LSRL to predict [rsp var] from [exp var] we are typically off by [value of s].

  • r2 (in %): About [r-sq]% of variation in [rsp var] is explained by the LSRL using [exp var].

  • Residual: The actual (rsp var) is about [residual] more/less than the predicted (rsp var).

Collecting Data

• Simple Random Sample (SRS): every group has an equal chance

• Cluster Sample: Divide pop. into heterogeneous groups [all from some]

• Stratified Random Sample: Divide pop. into strata of homogeneous groups [some from all]

• Stratifying: ↓ variability, ↑ precision

• Bias: undercoverage, nonresponse, response bias (inaccurate)

• EXPERIMENTS ASSIGN TREATMENTS

• Confounding: variable and the exp. variables affect the rsp. Variable and can’t be distinguished

• Experiments have comparison, random assignment, control, & replication

• Randomized block design: random assignment in each block

• Blocks share a var that may impact rsp, ↓ variability, allows for easier comparison

• Matched pairs: compare 2 treatments in block size 2

Probability, Random Variables, and Probability Distributions

• Probability: chance of an event occurring (0-1)

• $P(event) = \frac{successful}{total}$

• Complement: $P(not event) = 1 - P(event)$

• $P(A and B) = P(A∩B)$

• Conditional probability: $P(A) * P(B||A)$

• $P(A or B) = P(A U B) = P(A) + P(B) - P(A and B)$

• Conditional Probability: $P(A||B) = \frac{P(A and B)}{P(B)}$

• Mutually exclusive: $P(A or B) = P(A) + P(B)$

• Independent: $P(A||B) = P(A)$

• Random variables: quantitative, numerical values

• Discrete: only “whole” values possible

• Expected Value: $x1p1 + x2p2 + … + xnpn$

• Continuous: all values in an interval

• Binomial: Binary, Independent (10%), Number of fixed trials, Same probability of success p

• Geometric: Binary, Independent, same prob of success p, without trials set previously

Sampling Distributions

• Sample statistics estimate population parameters (x-bar, p-hat)

• Sampling distribution: distribution of values taken by a statistic in all possible samples

• Larger samples are less variable and more accurate

• Sampling distrib. of p-hat: approx normal if np >= 10 AND n(1-p) >= 10 (Large Counts Condition)

• Sampling distrib of x-bar: Check 10%/random condition, and n > 30

• Conditions: 10 of each, random, 10%

• Conf. Interval = Point Estimate +/- Margin of Error

• Larger sample size decreases margin of error!

• 1-sample CI: one-sample C% Z-Interval for p, hypothesis test = one sample z-test for p

• 2-sample CI: two-sample C% Z-Interval for p1 - p2, hypothesis test = two-sample z-test for p1 - p2

• NO paired data for props.

• State parameter(s) & hypotheses!

• 2-sided Ha: confidence interval to make a decision about H0. Reject if H0 not in interval [Conf level = opp of signif level)

• P-value: Assuming the [H0 in context] is true, the probability that the [observed statistic - x-bar or p-hat] will take a value as or more extreme than it does is [p-value].

• Confidence level: If we were to select many random samples and construct a C% confidence interval using each sample, about C% of the intervals would capture the [parameter in context].

• Confidence intervals: We are C% confident that the interval from ___ to ____ captures the [true parameter in context]

• Conditions - n > 30, random (no experiments), independent

• Confidence interval for one sample: one-sample C% t-interval for mu, hypothesis test = one-sample t-test for μ

• Confidence interval of two samples: 2-sample C% t-interval for μ1- μ2, hyp. test = two-sample t-test for μ1- μ2

• Paired data: experimental unit received 2 treatments. mu_diff

• CI = One-sample t C% CI for μdiff, Hyp test = One-sample t test for μdiff

• Type I Error: Rejecting H0 when we should’ve failed to reject H0

• Type II error: Failing to reject the H0 when we should’ve rejected H0

• Power = 1 - P(Type II error)

• To increase power: increase sample size, increase significance level, increase the diff b/w H0 and true Ha.

Chi-Square Inference

• Formula: 3 Tests: GoF, Independence, and Homogeneity

• Hypothesis Test

• Chi-Squared is a non-parametric test

• Calculate df (n-1)

• Conditions: random, indep., at least 5 success/fail

• Name the correct type of test

Inference for Slopes

• LSRL Equation: $μ = a + βx$

• Conditions: Linear, Indep, Normal, equal SD, Random

• Confidence interval: $b ± t*(SEb)$

• df=n-2

• Hypothesis test: $(\frac{B-β0}{SEb})$

Exploring One-Variable Data

• Categorical data: two-way tables & bar graphs, analyzing proportions

• Quantitative data: histograms, dotplots, box plots, stem and leaf plots, and scatterplots.

• Mean: non-resistant, Median: resistant

• Unimodal, Bimodal, Uniform

• Compare distributions using comparison words

• Histogram: approximate the mean and use words like “no more” / “approximately” when describing range

• Analyze distributions using CUSS in context: Center, Unusual features, Shape, Spread (skew pulls mean)

• Normal distribution: mound-shaped and symmetric $(\mu)$, $(\sigma)$

• Z-score: $\frac{value - mean}{SD}$

• Standard Normal Distribution: Mean of 0, SD of 1

• Empirical Rule: 68% (1 SD), 95% (2 SD), 99.7% (3 SD)

• Categorical data independence: conditional frequency = marginal frequency

Exploring Two-Variable Data

• Describe associations with direction, strength, form.

  • Direction: positive / negative (slope)

  • Form: linear / non-linear

• r (correlation coefficient): strength & direction, NOT FORM

• Least Squares regression line (LSRL): predicts values of response variable (y) given explanatory variable (x)

• LSRL: $ŷ = a + bx$

  • ŷ = predicted value of rsp variable

  • a = y-int, b = slope

• Residual = predicted - actual

• Random scatter on residual plot

• Extrapolation: predictions outside the interval of values of x

• S & R-sq influenced by outliers (s ↑, r-sq ↓)

• Interpretations:

  • Slope/b: As the [exp var.] increases by 1 [unit], the [rsp var.] is predicted to increase by b [units].

  • Y-intercept: When there are zero [exp var], the predicted [rsp var.] is y-int.

  • s: When using LSRL to predict [rsp var] from [exp var] we are typically off by [value of s].

  • r2 (in %): About [r-sq]% of variation in [rsp var] is explained by the LSRL using [exp var].

  • Residual: The actual (rsp var) is about [residual] more/less than the predicted (rsp var).

Collecting Data

• Simple Random Sample (SRS): every group has an equal chance

• Cluster Sample: Divide pop. into heterogeneous groups [all from some]

• Stratified Random Sample: Divide pop. into strata of homogeneous groups [some from all]

• Stratifying: ↓ variability, ↑ precision

• Bias: undercoverage, nonresponse, response bias (inaccurate)

• EXPERIMENTS ASSIGN TREATMENTS

• Confounding: variable and the exp. variables affect the rsp. Variable and can’t be distinguished

• Experiments have comparison, random assignment, control, & replication

• Randomized block design: random assignment in each block

• Blocks share a var that may impact rsp, ↓ variability, allows for easier comparison

• Matched pairs: compare 2 treatments in block size 2

Probability, Random Variables, and Probability Distributions

• Probability: chance of an event occurring (0-1)

• $P(event) = \frac{successful}{total}$

• Complement: $P(not event) = 1 - P(event)$

• $P(A and B) = P(A∩B)$

• Conditional probability: $P(A) * P(B||A)$

• $P(A or B) = P(A U B) = P(A) + P(B) - P(A and B)$

• Conditional Probability: $P(A||B) = \frac{P(A and B)}{P(B)}$

• Mutually exclusive: $P(A or B) = P(A) + P(B)$

• Independent: $P(A||B) = P(A)$

• Random variables: quantitative, numerical values

• Discrete: only “whole” values possible

• Expected Value: $x1p1 + x2p2 + … + xnpn$

• Continuous: all values in an interval

• Binomial: Binary, Independent (10%), Number of fixed trials, Same probability of success p

• Geometric: Binary, Independent, same prob of success p, without trials set previously

Sampling Distributions

• Sample statistics estimate population parameters (x-bar, p-hat)

• Sampling distribution: distribution of values taken by a statistic in all possible samples

• Larger samples are less variable and more accurate

• Sampling distrib. of p-hat: approx normal if np >= 10 AND n(1-p) >= 10 (Large Counts Condition)

• Sampling distrib of x-bar: Check 10%/random condition, and n > 30

• Conditions: 10 of each, random, 10%

• Conf. Interval = Point Estimate +/- Margin of Error

• Larger sample size decreases margin of error!

• 1-sample CI: one-sample C% Z-Interval for p, hypothesis test = one sample z-test for p

• 2-sample CI: two-sample C% Z-Interval for p1 - p2, hypothesis test = two-sample z-test for p1 - p2

• NO paired data for props.

• State parameter(s) & hypotheses!

• 2-sided Ha: confidence interval to make a decision about H0. Reject if H0 not in interval [Conf level = opp of signif level)

• P-value: Assuming the [H0 in context] is true, the probability that the [observed statistic - x-bar or p-hat] will take a value as or more extreme than it does is [p-value].

• Confidence level: If we were to select many random samples and construct a C% confidence interval using each sample, about C% of the intervals would capture the [parameter in context].

• Confidence intervals: We are C% confident that the interval from ___ to ____ captures the [true parameter in context]

• Conditions - n > 30, random (no experiments), independent

• Confidence interval for one sample: one-sample C% t-interval for mu, hypothesis test = one-sample t-test for μ

• Confidence interval of two samples: 2-sample C% t-interval for μ1- μ2, hyp. test = two-sample t-test for μ1- μ2

• Paired data: experimental unit received 2 treatments. mu_diff

• CI = One-sample t C% CI for μdiff, Hyp test = One-sample t test for μdiff

• Type I Error: Rejecting H0 when we should’ve failed to reject H0

• Type II error: Failing to reject the H0 when we should’ve rejected H0

• Power = 1 - P(Type II error)

• To increase power: increase sample size, increase significance level, increase the diff b/w H0 and true Ha.

Chi-Square Inference

• Formula: 3 Tests: GoF, Independence, and Homogeneity

• Hypothesis Test

• Chi-Squared is a non-parametric test

• Calculate df (n-1)

• Conditions: random, indep., at least 5 success/fail

• Name the correct type of test

Inference for Slopes

• LSRL Equation: $μ = a + βx$

• Conditions: Linear, Indep, Normal, equal SD, Random

• Confidence interval: $b ± t*(SEb)$

• df=n-2

• Hypothesis test: $(\frac{B-β0}{SEb})$