STAT Exam 2 Review TOPIC 5 Sampling Distributions Sample mean X good estimate of the population mean o Means 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 Population distribution mean u standard deviation Random sample sample size n sample mean x Sampling distribution of the sample mean Mean of x Standard deviation of x o Variance 2 n When 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 2 n o Central 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 2 n Sampling distribution of sample proportion percentage o 2 values success and failure o P proportion of individuals in the population whose variable value is success o Choose random sample size from large pop that contains p o p sample proportion of successes Bernoulli 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 bino m n p p x n or x np o o o Mean up p Variance 2 p p True for all n As sample size increases sampling distribution of p becomes approx normal If 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 distribution When np 10 OR n 1 p 10 p does not have normal distribution TOPIC 6 Inferences about a Single Mean Population mean population proportion p Parameters of sampling distribution 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 of normal population S D known use z procedure n 30 n 30 normal distribution If n 30 use population S D if known or sample S D population S D is not given use z procedure unknown mean n random sample size C z m x sample mean N m z S D sqrt n margin of error S D unknown use z procedure large sample size n 30 unknown mean X point estimate S sample S D s sqrt n standard error Pop S D unknown use t procedure small sample size n 30 normal population distribution Draw random sample of size n from NORMAL population unknown mean X point estimate S sample S D s sqrt n standard error If population S D is not known calculate sample S D use t procedure t score value t from t distribution with degrees of freedom n 1 As n increases t distribution approaches the standard normal distribution Confidence intervals for p z procedure p unknown proportion p sample proportion Level C CI for p If working with proportions success rates Use z procedure for p Level C CI for u Sample size with margin of error given Rounded UP to the next integer of successes in sample of failures in sample use z procedure for p Sample size calculations Draw a random sample of size n from a normal population or a large random sample from a non normal pop u unknown mean known S D Draw a random sample of size n from a population point estimate m As n increases m decreases Level C CI for p Sample size needed to get a specified m having p unknown proportion p 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 situation Sampling design describes how to choose a sample from the population Sampling frame list of individuals from which a sample is actually selected o Under 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 random sample of individuals without the condition controls o Medical and lifestyle histories of subjects in each group to learn what factors may be associated with the condition Cohort study prospective subjects share a common demographic characteristic are enrolled and observed at regular intervals over extended time o Start with one homogeneous group o Provide info about relative health risks of different subgroups 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 at individuals in order to observe their responses purpose is to study whether treatment causes a change in the response random 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 twice o Require 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 block o Block group known before experiment to be similar in some way that is expected to affect the response to the treatments TOPIC 8 Hypothesis Tests Checks sample data against a claim or assumption about the population parameter Hypothesis tests for u o When is known z test n 30 OR n 30 normal distribution o When is not known one sample t test n 30 normal population distribution Steps of hypothesis testing Null hypothesis H0 the claim being tested Set