of Probability an event is its long-run relative frequency; Must be legitimate 0 P 1 & sum of set of P's = 1 Event combination of outcomes For any random phenomenon, each attempt (or trial) generates an outcome (Discrete distinct values / Continuous some range) Something Has to Happen Rule the probability of the set of all possible outcomes of a trial = 1 P(S) = 1 S is set of all possible outcomes Independent- Formal Def- Events independent if P (B|A) = P(B) prob of independent events don't change when you find out one of them has occurred * No such thing as law of averages (promises short-run compensation for recent deviations from expected behavior) Law of Large Numbers long-run relative frequency of repeated independent events gets closer to true relative frequency as number of trials increases Disjoint (Mutually Exclusive) events that can't occur together; no outcomes in common Complement Rule The probability of an event occurring is 1 minus the probability that it doesn't occur P(A) = 1 P (AC) Multiplication Rule for INDEPENDENT events A & B P (both A & B) = P(A) x P(B) "Some" means at least one complement of this is none; so P(at least one A in 5 trials) [ P(A) = .35 / P (AC) = .65; P(none) in all 5 = (.65) ^ 5 Complement = 1 ((.65) ^5) Sample Space collection of all possible outcomes (which don't have to be equally likely); Ex. If pulling bills from wallet S = {1, 5, 10, 20, 50, 100} General Addition Rule DO NOT need disjoint events; P (A or B) = P (A) + P(B) P (A & B) General Multiplication Rule DO NOT need independence; P( A and B) = P (A) * P(B |A) OR P(B) * P(A|B) Conditional Probability P ( B | A) = prob of B given A = P(A & B) / P(A) *Mutually Exclusive events CANNOT be Independent -knowing one occurred means other didn't (dependent); if P (A & B) exists, not mutually exclusive Venn Diagram: Tree Diagram:
Drawing w/o replacement denominator changes because one less available each drawing; Ex: drawing something randomly like rooms ** P(B|A) P(A|B) [Baye's Rule reverses conditional probs] Model for Distribution of Sample Proportions How a statistic (phat, b1, ybar) is distributed, NOT how data (p, , ) is distributed -Randomization leads to sample-to-sample variationcreates sampling distribution for samples of particular size n -Distribution is N (p, (p*q / n)) Normal model centered at true p with SD of (p*q / n); Claim is more true as n grows... Assumptions: 1) Sampled values independent of each other 2) Sample size (n must be large enough) Conditions: 1)10 % Condition sample size, n, must be no larger than 10% of population (if sample made w/o replacement) 2) Success/Failure Condition the number of successes (n*p) and number of failures (n*q) must both be greater than 10 Central Limit Theorem the sampling distribution model of the sample mean & proportion is approximately Normal for large n, regardless of the distribution of the population, as long as observations are independent; Normal model has mean (ybar) equal to population mean and standard deviation (ybar) = SD(ybar) = / (n) **Can't use it with individuals or single events--MUST BE GROUPS
-Doesn't talk about distribution of data within population--talks about means of different samples drawn from same population (a sampling distribution model); if population skewed & samples small, means won't be Normally distributed -If very skewed population, sample must be very large for Normal model to work / If population nearly Normal, even small samples will work -Assumptions & Conditions: 1) Random Sampling Condition- data values must be sampled randomly 2) Independence Assumption- no way to check in general, but when sample drawn w/o replacement 10% Condition- n no more than 10% of population -As samples get larger, we expect data to look more like pop. from which it was drawn--skewed, bimodal, whatever, NOT necessarily Normal -Means vary less than individual observations (they have smaller SDs than individuals)--Sample Mean is a random variableadditional SRS will have different means Diminishing Returns SD of sampling distribution declines only w/ square root of sample size Standard Error- whenever we estimate the standard deviation of a sampling distribution using statistics from the Data; s instead of / phat instead of p Sampling Distribution Models 2 Basic Truths: 1) Sampling distributions arise because samples vary (each random sample contains different cases and yields different statistics) 2) Although we can always simulate a sampling distribution, CLT saves us the trouble for means and proportions
GENERAL CI: Margin of Error- the extent of the interval on either side of phat, generally a confidence interval = estimate ME CI / 2 = ME -The MORE confident, the LARGER the Margin of Error---meaning, more confidence (certainty) makes for lower precision, and vice versa Critical Value the number of SE's used to give the confidence interval (If using Normal model z* If using Student's tt*df) A Better CI for Proportions- make interval more accurate by adding 4 phony observations (2 success/ 2 failure) so new p = y+2 / n+4, then make CI 90% Confident = 90% of all random samples will have mean/proportions/associations that fall within the CI (a,b) GENERAL HYPOTHESIS TESTS: Null Hypothesis- H0 usually a statement of no change from traditional value or no affect or no difference ***USE SD () NOT SE- because we have not estimated anything; SD(phat) comes from H0, then we must find how likely it is to see observed value **H0 and HA true value of pop parameter (model) not statistic c& only 1 value; Use what you want to show as HA, so if reject H0 left with that P-value- the probability that the observed statistic value, or more extreme, could occur if the null model were true; if P-value small enough, we reject H0 -P-value is a CONDITIONAL PROB: P (observed statistic [or more extreme] | H0 true)--it's NOT the prob that H0 is true Alpha Level (Significance Level) changes depending on situation; when we reject H0 significant at the x % significance level -Reject H0 (statistically significant) P-value < -Retain H0 (not stat significant) P-value > Type I Error- - Mistakenly reject a TRUE H0 (False Positive); same prob as HIGHER = HIGHER chance for Type I Error Type II Error- - Fail to reject a FALSE H0 (False Negative); more difficult to calculate since HA gives range of possible values **ONLY way to reduce BOTH errors is to reduce SD (increase n- dimin returns); otherwise decrease one by increasing otherinverse/trade-off Power- ability to REJECT a FALSE H0; Power = 1 ; when we calculate power, must imagine H0 is false; DECREASE = DECREASE power Effect Size distance between H0 value (p0) and truth (p); Power depends on effect size easier to see larger effects (higher power) -The larger the difference between hypothesized p0 and true p the SMALLER the and the LARGER the Path for Hypothesis Testing: 1. Hypothesis: H0 and HA 2. Plan: state model to be used; Assumptions/Conditions; include name of test 3. Mechanics: calculations of test statistic, goal is to obtain P-value 4. Conclusion: whether we reject or fail to reject H0 One-Tailed: when HA focuses on deviations in only one direction (HA: p > p0 or p > p0); P-value is the probability of deviating only in the direction of the alternative away from the null hypothesis value; more, less, increase, decrease" Two-Tailed: when we are equally interested in deviations on either side of H0 (HA: p p0); P-value is the probability of deviating in either direction from the null hypothesis valuemust ADD the probabilities in both tails of the sampling distribution model; "different" ***DON"T INTERPRET SIGNIFICANT AS CAUSAL ***STAT SIGNIFICANT DOES NOT MEAN "ACTUALLY IMPORTANT" **T-model compensates for extra variability due to fact that SE is based on s, which varies from sample to sample in addition to variation of ybar ** Larger Sample Size Smaller ME Greater Precision **ON T-table--if n is in between 2 df values, use LOWER one BOTH: **Any H0 that falls within CI, we retain; Any H0 that falls outside CI, we reject **Z-score = observed actual / SD In general, a CI with a confidence level C% corresponds to 2-sided hypothesis test with level 100 - C% ** 1-sided hypothesis with test level .5*(100-C)% **Statistically Significant = NOT a result of natural sampling variation alone
One Proportion Z- Interval: CI takes a sample & gives region where TRUE value for a proportion lies CI is phat z* x SE(phat) We are 95% confident that the true proportion lies between (z,y) Solve for sample size: n = [(z* ^2) pq ] / ME^2 One Proportion Z-Test: Is your sample statistically significant (different)? One or two-sides H0 z = (phat p0) / SD (phat) Assumptions: 1) Independence ( data values are independent) 2) Sample is sufficiently large (since model used for inference is based on CLT) Conditions: Independence A. Plausible Independence (does it make sense) B. Randomization (SRS or properly randomized experiment C. 10% Condition (if w/o replacement, sample must represent less than 10% of population) Sample Size A. Success/Failure Condition (at least 10 successes and 10 failures) Watch OutCI: 1. Don't claim other samples will agree with yours 2. Don't be certain about the parameter 3. Don't forget it's the parameter, not the phat 4. Don't claim to know too much (talk about population sampled from) 5. Take Responsibility 6. Watch out for biased samples 7. Don't suggest the parameter varies (by saying there is a 95% chance...) 8. Think about independence Watch OutHypothesis: 1. Don't make H0 what you want to show 2. Don't confuse practical and statistical significance 3. Might still make a wrong decision (Type I and Type II error never both 0) 4. Always CHECK CONDITIONS 5. Don't believe too strongly in arbitrary ***Need LARGER samples to estimate proportions near 0% or 100% Two Proportion Z-Interval: (phat1 phat2) z* x SE (phat1 phat2); z* depends on particular confidence level, C, that is specified; We are 95% confident that the proportion of sample1 is between a & b higher than the proportion of sample2 Two Proportion Z-Test: usually H0 says there is no true difference p1 p2 = 0 z = (phat1 phat2) (0) / SEpooled(phat1 phat2) SO, VARIANCES are EQUAL we can POOL (POOLING GETS MORE ACCURATE VALUE) -Pooling- combining counts to get an overall proportion(when sources homogeneous) ; only when hypothesis says true proportions are equal -Can pool because proportions and SD are linked -phat (pooled) = success1 + success2 / n1 +n2 -When we have proportions and not counts, multiply phat1 by n1 and phat2 by n2; if not whole ROUND DOWN to nearest whole Assumptions: 1) Samples are independent 2) Data values in each sample are independent 3) Both samples sufficiently large Conditions: Independence A. Randomization (think about if data drawn independently and at random) B. 10% Condition (w/o replacement, data less than 10% of population) C. Independent Samples (can't be related) Sample Size A. Success/Failure Condition for Each Sample Watch Out Both: 1. Don't use these methods when samples aren't independent or non-random 2. Don't interpret significant difference causally ***The variance of the sum or difference of two INDEPENDENT RANDOM variables is the SUM OF THEIR VARIANCES **For means, if you know (SD) use z (that's rare), but if you use s to estimate (SE), use t-model *** Use boxplots to compare the two groups One Sample Mean T-interval: CI is ybar t*(df = n 1) x SE(ybar); We are 95% confident that the true mean speed of all vehicles is btwn a & b One Sample Mean T-test: Usually test hypothesis H0: = 0 t = (ybar ) / SE (ybar) Assumptions: 1) Independence Assumption 2) Normal Population Assumption (to use Student's t, data must be from pop that follows Normal model) Conditions: Independence A. Randomization (random sample / randomized experiment) B. 10% Condition (important if large pop or small sample) Normal Population A. Nearly Normal Condition (data comes from distribution that is unimodal / symmetric; check histogram or Normal Probability Plot (should be straight line to be Normal) n < 15 if any outliers / strong skewness, do NOT use these methods / 15 <n <40 unimodal & reasonably symmetric / 40 < n skewed is OK Watch Out Both: 1. Beware of skewed data 2. Beware of multimodality 3. Set outliers aside 4. Watch out for bias 5. Independent Data 6. Make sure sample appropriately randomized Two Independent Samples Mean T-test: H0: 1 2 = 0 usually 0=0 t = [ (ybar1 ybar2) (0) ] / SE(ybar1 ybar2) Two Independent Samples Mean T-interval: (ybar1 ybar2) t* x SE (ybar1 ybar2) df for t* needs special formula Assumptions / Conditions: Same as One Sample + 1 more Assumption: 1) Independent Groups Assumption (think about how data collected) If Variances are EQUAL (as in H0)Pooled t-test: But Means don't have a link between their value and their variance--SO we must be willing to assume -Equal Variance Assumption (rest of conditions same as 2 sample t-test) Watch Out Both: 1. Watch out for paired data (like before & after) 2. Look at plots (boxplots) for outliers and non-Normal distributions Matched Pairs Mean Mechanically same as 1-sample t-test for the means of the pairwise differences Paired Data when observations are collected in pairs or the observations in one group naturally related to observations in another (violates independence) -Look for pairwise differences, and treat the differences as if they were the data--controls variability between individuals by just looking at pairwise Assumptions: 1) Paired Data Assumption 2) Independence Assumption (if data paired, groups are NOT independent, but the differences must be independent of each other 3) Normal Population Assumption (population of differences must follow a Normal model) Conditions: Independence A. Randomization Condition (individuals must be random sample or treatments in experiment must be randomly assigned) Normal Population A. Nearly Normal Condition (check w/ histogram or Normal probability plot--even if original measurement skewed or bimodal, differences may be nearly Normal) n= number of pairs Paired T-test for Mean Difference between 2 groups: H0: d = 0; d = pairwise differences and 0 usually 0 t(df=n 1) = (dbar - 0) / SE(dbar) Paired T-Interval: CI = dbar t* (df=n 1) x SE(dbar) **Comparing Boxplots in Paired test shows NOTHING it's the differences we care about Watch Out Both: 1. Outliers within differences 2. Don't look for difference in side-by-side boxplots 3. Don't use paired method when not paired Idealized Regression Line: written w/ Greek letters and consider coefficients to be PARAMETERS; 0 = intercept 1 = slope Corresponding to out fitted line = b0 + b1x we write y = 0 + 1x y instead of because this is a MODEL Not all y's are at these means (some above/below), so like all other models, this model makes errors = y y -To talk about individual y's instead of means y = 0 + 1x + Residuals (e = y ) are the sample based versions of the errors Assumptions: 1) Linearity Assumption 2) Independence Assumption (errors in underlying regression model must be independent 3) Equal Variance Assumption (variability of y should be same for all x values) 4) Normality assumption (errors around idealized regression line at each value of x follow a Normal model--to use Student's t) Conditions: Linearity A. Straight Enough Condition (scatterplot looks straight / Residuals vs. should have no pattern) IndependenceRandomization Condition (individuals are representative sample of population (Residual vs X-values plot should have no pattern) Equal Variance Does the Plot Thicken? Condition (check X vs Y scatterplot or Residuals vs x (or y) plot--do NOT want it to thicken) Normality Nearly Normal Condition (check histogram or Normal probability plot of Residuals--less important as sample size grows because model about means and CLT takes over) ***If all 4 assumptions Good, idealized regression should have Normal distribution of y-values @ EACH x-value Spread around the line is measured with the residual SD (se), often labeled just s ++Use of Residuals to estimate SD is a form of pooling 3 Aspects of scatterplot affect the SE of regression: 1) spread around the line (se the smaller se (less scatter) the stronger the relationship btwn x & y) 2) Spread of x values(sx)broader range = more stable (precise) base for slope 3) Sample Size (n) larger n (more dots) gives more stability SE (b1) = se / ((n-1) * sx) se increases SE (b1) increases n or sx increases SE (b1) decreases