08 11 2010 11 02 00 Random variable a numerical measurement of the outcome of a random process finite number of values Discrete random variable a variable that can assume only a countable Number of tvs in a household can t have 2 5 variable can take on and their probabilities Probabilities will add up to 1 Probability distribution function specifies the possible values a random Variance of a random variable is a weighted avg of the squared deviations x M 2 each outcome is weighted by its probability in order to take into account outcomes that are not equally likely the variance the more scattered the values of x on average Standard deviation of x the positive square root of the variance the larger Binomial model trial has only two possible outcomes success fail there is a fixed number n of identical trials the trials are independent of each other the probability of success p remains constant from trial to trial if p is the probability of success 1 p q is probability of failure o example manufacturer labels items defective or acceptable o job applicants either accept or reject offer Normal Distribution probability is measured from the area under the curve o total area 1 Standard Normal Distribution z distribution mean 0 SD 1 values above the mean have positive z values values below the mean have negative z values Statistical Experiment data are fixed experiment is random many possible samples Survey every group of 1000 people is equally likely Experiment each assignment to control or treatment could happen Sampling variability each time we take a random sample from a population we are likely to get a different set of individuals and calculate a different statistic Sampling distribution the variation from sample to sample follows a predictable pattern when we take a lot of random samples of the same size from a given population Distribution of the statistic obtained from repeated samples using the Normal approximation is most accurate for a large fixed n when p is close to Rules for Sample Proportion a binomial experiment with normal same number of observations Changes with parameter 0 5 and least accurate when p is near 0 or near 1 approximation A population with a fixed proportion p Random Sample independent equal chance Sample size is large np 9 n 1 p 9 Empirical rule 68 between M sd and M sd 95 between M 2 sd and M 2 sd 99 7 between M 3 sd and M 3 sd 95 confidence We are 95 confident that between and of people in Washington agree with the recent changes to bankruptcy laws 95 of samples of this size will produce confidence intervals that capture we expect 5 of our samples to produce intervals that fail to capture the the true proportion true proportion 95 of samples intersect the true proportion capture true proportion Want confidence as high as possible but makes interval wider Lower confidence interval becomes narrower Assumptions and Conditions Increasing sample size makes Confidence Interval narrower To reduce margin of error use lower confidence level Independence assumption are sample observations independent of each other Randomization condition was the sample randomly generated 10 condition if sampling is done without replacement then the sample size n must be no larger than 10 of the population Success failure condition the sample size must be large enough so that both np and n 1 p are at least 10 Stating Hypotheses Two explanations o The effect is due to chance variation o The effect is due to something significant How to decide o Null hypothesis HO the coin is fair o Alternative hypothesis Ha the coin is not fair population population conclude that Null hypothesis a very specific statement about a parameter of the Labeled Ho states status quo previous knowledge no effect The one that we want to reject Alternative hypothesis a more general statement about a parameter of the the opposite of the null Labeled Ha The one we try to prove Ho the defendant is innocent If sufficient evidence is presented the jury will reject this hypothesis and Ha the defendant is guilty Two tail or two sided test of the population proportion has Null Ho p po a specific proportion Alternative Ha p not po One tail or one sided test of a population proportion Null Ho p po a specific proportion Alternative Ha p po OR p po P value tests of statistical significance quantify the chance of obtaining a particular random sample result if the null hypothesis were true a way of assessing the believability of the null hypothesis given the evidence provided by a random sample Small p value implies that random variation due to the sampling process alone is not likely to account for the observed difference o Reject Ho the true property of the population is significantly different from what was stated o Strong evidence AGAINST Ho o Usually a p value of 0 05 or less is considered significant the phenomenon observed is unlikely to be entirely due to chance event from random sampling Test for a population proportion If Ho is true the sampling distribution is known the likelihood of our sample proportion given the null hypothesis depends on how far from po our p hat is in units of standard deviation Four Steps of hypothesis testing Define the hypothesis to test and the required significance level Calculate the value of the test statistic State the conclusion Find the p value based on the observed data o Reject null hypothesis if p value sign level o If p value sign level data does not provide sufficient evidence to reject the null hypothesis how much evidence against Ho we require Significance level the largest p value tolerated for rejecting a true null Decided arbitrarily based on random choice or personal whim BEFORE conducting the test o If p value is equal to or less than sign level we REJECT Ho o If p value is greater than sign level then we FAIL TO REJECT Ho Rejection region shaded area on the tail side When z score falls within it p value is smaller than sign level and you have shown statistical significance Rejection region for a two tail test of p with sign level 0 05 5 Two sided test means that sign level is spread between both tails of the curve o Middle area C of 1 sign level 95 o Upper tail area of sign level 2 0 025 Statistical significance vs practical significance Statistical significance only says whether the effect observed is likely to be due to chance alone because of random sampling o May not be practically important o Doesn t tell you about the magnitude of the effect only that there is
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