STAT Exam 2 Review TOPIC 5 - Sampling Distributions-Sample mean, X, good estimate of the population mean, µoMeans of random samples are less variable/more normal than individual observations-Law of Large Numbers - the larger the sample, the statistic sample mean, X, gets closer to the population mean µ-Population distribution: mean u , standard deviation σ-Random sample: sample size n, sample mean x_-Sampling distribution of the sample mean, µMean of: µx Standard deviation of: σxVariance = σ2/n oWhen the population distribution is normal, N(, 2),-The distribution of sample mean is also normal distribution with mean u and variance-X has normal distribution when X_~ N(µ, σ2/n)oCentral Limit Theorem (CLT) - When the population distribution is NOT normal-Larger sample sizes converge to normal distribution with mean u and variance-Sample size of AT LEAST 30 is required to use CLT, X_~ N(µ, σ2/n)-Sampling distribution of sample proportion/percentageo2 values = success and failureoP = proportion of individuals in the population whose variable value is "success"oChoose random sample size from large pop. that contains pop^ = sample proportion of successesBernoulli trial-2 values = success and failure-The probability of success, P(success) = p for all trials-X counts number of successes in n independent trials-X~binom(n,p)-p^ = x/n or x = np^Parameters of sampling distribution-Mean - up = p-Variance - σ2 pσp = True for all noAs sample size increases, sampling distribution of p^ becomes approx. normaloIf np ≥ 10 and n(1-p) ≥ 10, then the sampling distribution of p^ is approx normal, N(p, p(1-p)/n)-p^ has normal distributionoWhen np < 10 OR n(1 - p) < 10, p^ does not have normal distribution TOPIC 6 - Inferences about a Single Mean -Population mean µ, population proportion p-Point estimate - single best guess of the unknown parameter-Confidence interval - interval of possible values of the unknown parameter-Hypothesis test (test of significance) - assessment of the evidence for or against a claim about the unknown parameter Confidence intervals for µ of normal population S.D. knownuse z-procedure(n ≥ 30)(n < 30) + normal distribution µ - unknown meann - random sample sizex - sample meanz* (S.D/sqrt(n)) - margin of error C ^, z* ^, m ^N ^, m \/If n≥30, use population S.D (if known) or sample S.D (population S.D is not given), use z-procedure S.D. unknownuse z-procedurelarge sample size (n > 30) µ - unknown meanX - point estimateS - sample S.D.(s/sqrt(n)) - standard error Pop. S.D. unknownuse t-proceduresmall sample size (n < 30) + normal population distribution Draw random sample of size n from NORMAL populationµ - unknown meanX - point estimateS - sample S.D.(s/sqrt(n)) - standard error t-score: value t* from t-distribution with degrees of freedom (n-1)-As n increases, t-distribution approaches the standard normal distributionIf population S.D is not known, calculate sample S.D, use t-procedure Confidence intervals for p z-procedure p - unknown proportionp^ - sample proportion # of successes in sample# of failures in sample >use z-procedure for pLevel C CI for p If working with proportions (success rates)-Use z-procedure for pSample size calculationsDraw a random sample of size n from a normal population (or a large random sample from a non-normal pop) u - unknown meanσ - known S.DLevel C CI for upoint estimate +- (m)-As n increases, m decreasesSample size with margin of error given-Rounded UP to the next integer Draw a random sample of size n from a population Level C CI for p Sample size needed to get a specified mhaving p - unknown proportionp^ - sample proportion -Use given p*-If p* is not given, p* = 0.5-Rounded UP to the next integer TOPIC 7 - Collecting Data-Population - the entire group of individuals about which we want information-Sample - part of population from which we collect information or perform an experiment on-Explanatory variable - independent, x-Response variable - dependent, y-Lurking variable Observational studies - (surveys and sampling) OBSERVES individuals and measures variables of interest but does not attempt to influence the responses; purpose is to describe some group/situationSampling design - describes how to choose a sample from the population-Sampling frame - list of individuals from which a sample is actually selectedoUnder-coverage - frame that leaves out part of the population-Simple random sample (SRS) - each possible sample of that size has the same chance of being selected-Stratified sample - divides the population into separate groups formed naturally, called strata, then selects SRS of subjects from each stratum Comparative observational studies-Case control study (retrospective) - random sample of individuals with a condition (cases) is compared with a randomsample of individuals without the condition (controls)oMedical and lifestyle histories of subjects in each group to learn what factors may be associated with thecondition-Cohort study (prospective) - subjects share a common demographic characteristic are enrolled and observed at regular intervals over extended timeoStart with one homogeneous groupoProvide info about relative health risks of differentsubgroups Sources of bias-Under-coverage - occurs when some groups in the population are left out of the process of choosing the sample (homeless are left out of surveys that sample households)-Nonresponse bias - serious source of bias in most sample surveys and occurs when a selected individual cannot be contacted or refuses to participate-Response bias - occurs when a subject gives incorrect response or question wording is confusing/misleading Experiments - (clinical trials) IMPOSES treatment on Types of experimental designs-Complete randomized design - all subjects are allocated atindividuals in order to observe their responses; purpose is to study whether treatment causes a change in the responserandom among all the treatments-Matched pairs design - compares EXACTLY two treatments, using a series of individuals that are closely matched two by two or by using each individual twiceoRequire that the assignment of two treatments within each pair be randomized to avoid systematic bias-Randomized block design - random assignment of individuals to treatments is carried out separately within each blockoBlock - group known before experiment to be similar in some way that is expected to affect the response to the treatments TOPIC 8 -