Chapter 8 Random Variable a variable with its value based on the outcome of a o Discrete Random Variable when you can list all the possible random event outcomes o Continuous Random Variable a random variable that can take on any value between two values Probability Model for both discrete and continuous variables the collection of all possible values and the probabilities with them o It is written as die example P X x 1 6 if x 1 2 3 4 5 6 Expected Value E x or of a discrete random variable x P x 2 Var X x EV Rules for means and variances 2P X E x2 E x 2 2P X x o E a bX a bE X o Var a bX b2Var X o SD aX a SD X Addition rule for expected values of random variables E X Y E X E Y Addition rule for variances of independent random variables Var X Y Var X Var Y SD 2X 2SD X Discrete Uniform Distribution when X is a random variable with possible outcomes 1 2 n and P X i 1 n for each value of i Binomial Probability Model defined by the number of trials n and the probability of success p o Denoted as Binom n p Conditions N independent trials 2 outcomes for each trial P S is the same for all trials Must use combinations to find this n The probability of getting exactly k successes in n trials is k nCk n k n k P x k n k pkqn k np npq When using normal model np 10 and nq 10 Continuity Correction when using a normal model for a discrete variable going up and down 5 to consider what is included o Ex P x 10 with a normal model is P x 9 5 o Ex P x 10 should not include 10 so do P x 9 5 o Ex P x 10 with a normal model is P x 9 5 o Ex P x 10 with a normal model is P x 10 5 Probability Density Function PDF shows the distribution of the probability for any continuous random variable Requirements always nonnegative and total area 1 0 Uniform F x 1 b a if a x b P c X d d c b a E X a b 2 Var X b a 2 12 Normal Model distribution must be unimodal and roughly symmetric N Standard Normal Model scores 0 and 1 normally with z 68 95 99 7 Rule Empirical Rule tells us roughly how the values are distributed Look an tables of normal percentiles using standard deviations Sum or difference of two independent normal random variables is also normal Success Failure Condition a binomial model is approximately normal if we expect at least 10 successes and 10 failures o np 10 and nq 10 Sample must be less than 10 of population Bernouli Trials Success or fail p success q 1 p failure trials are independent Geometric Model How many trials till first success Geom p P X x qt 1p EV 1 p q p2 Poisson Model mean x number of occurrences P X x e x x E x SD x Calculator Stuff 2nd vars Binom PDF n p x Normal CDF start stop Poisson PDF x For all other stuff is is under stat tests Chapter 9 Population mean Sample statistic xbar Population proportion p Sample proportion p Sampling Distribution the distribution of proportions over many independent samples from the same population all p s o Sampling distributions of proportions categorical values p not p SD pq n Assumptions Conditions Independence assumption Sample size assumption sample size is large enough Randomization Condition sample must be representative of population 10 Condition sample size n must be less than 10 of population Success Failure Condition np 10 and nq 10 o Sampling distributions of means Central Limit Theorem the mean of a random sample has a sampling distribution whose shape can be approximated by a normal model The larger the sample the better the approximation will be The sampling distribution of any mean becomes normal as the sample size grows regardless of the shape of the population distribution Law of Diminishing Returns the idea that sampling n times more people only reduces the standard deviation to n Conditions o Independence o Randomization Confidence Intervals o Standard error standard deviation using p SE p q n o Use a normal model centered at p and us SE instead of SD o we are 95 confident that the parameter was captured in the interval o One proportion z interval p z SE o Margin of error the extend of that interval on either side of p Balance between certainty and precision ME z p q n o Critical Value z the number of SE s we must stretch out on either side of p 90 z 1 645 95 z 1 96 99 z 2 576 o Conclusion with X confidence the proportion of people in the population of is between Do Not Say the probability that the proportion of people in the population is between o Assumptions Conditions Independence assumption Randomization condition 10 condition sample must be less than 10 of population success failure condition np 10 and nq 10 o When using margin of error to calculate sample size always round up Sampling Error Sampling Variability not really an error rather the variability you d expect to see from one sample to another The normal model is more and more applicable to sampling distributions as the sampling size gets larger Law of Large Numbers as the sample size gets larger each sample average tends to become closer to the population mean o Also becomes more and more bell shaped like a normal model Chapter 10 Null Hypothesis H0 specifies the population model parameter and proposes a value for that parameter o H0 p p0 Alternative Hypothesis HA contains the values of the parameter that we consider plausible if we reject the null hypothesis o p p0 or p p0 or p p0 When conducting a hypothesis test using the rejection region basically just use alpha and calculate the minimum z value Use the mean and standard deviation form the null hypothesis population o SD p p0q0 n P value the value on which we base our decision One Proportion z test Low P values reject the null hypothesis o The conditions for the one proportion z test are the same as for the one proportion z interval We test hypothesis H0 p p0 using the statistic Z p p0 SD p Two sided alternative when we are equally interested in deviations on either side of the null hypothesis value One sided alternative an alternative hypothesis that focuses on deviations from the null hypothesis value in only one direction Alpha Level the threshold at which results are statistically significant o Also called confidence level o Statistical significance only says whether the effect observed is likely to be due to chance alone because of random sampling The p value does not show the magnitude of the effect …
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