CHAPTER 8 RANDOM VARIABLES AND PROBABILITY MODELS Discrete Random Variable Discrete random variable Random variable is a numerical measurement of the outcome of a random process a variable that can assume only a countable finite number of values 1 3 8 2 3 8 Probability distribution function pdf specifies the possible values a random variable can take on and their probabilities 0 k P X k 1 8 1 Probabilities add up to 1 2 0 P X k 1 for every k 3 Other number have prob 0 Expected value of a discrete random variable Mean over the long run expected value X is found by multiplying each possible value of X by its probability and then adding the products i e E X k P X k Greek Lesson E X population mean 2 Var X st dev population variance For a discrete random variable X with probability distribution and mean X the variance 2 of X is found by multiplying each squared deviation of X by its probability and then adding all the products In general Var X E X 2 E X2 2 Var X Y Var 2X 3 1 8 Rules for Means and Variances If X is a random variable and a and b are fixed numbers then E a bX a bE X Var a bX b2Var X If X and Y are two independent random variables then E X Y E X E Y Var X Y Var X Var Y Var X Y Var X Var Y The standard deviation of a random variable is SD X sqrt Var X The Binomial Distribution 1 A trial has only two possible outcomes success or failure 2 There is a fixed number n finite of identical trials 3 The trials of the experiment are independent of each other 4 The probability of a success p remains constant from trial to trial 5 If p represents the probability of a success then 1 p q is the probability of a failure Binomial coefficient n choose k Bionomial Distribution Formula P x probability of x successes in n trials with probability of success p on each trial x number of successes in sample x 0 1 2 n p probability of success per trial q probability of failure 1 p n number of trials sample size Summaries of Binom Dist E x np 2 Var X npq and sqrt npq The Normal Distribution Bell Shaped Symmetrical Mean Median Mode Mean St Dev Random Variable has theoretical range infinity to infinity By varying we obtain different normal distributions One st dev 68 two 95 three 99 7 Probability of Compliment P Ac 1 P A np nq 10 CH 9 SAMPLING DISTRIBUTIONS AND CONFIDENCE INTERVALS FOR PROPORTIONS Statistical Experiment statistics calculated x bar s parameters unknown population proportion p Rules for Sample Proportion 1 pop with fixed proportion p 2 random sample independent equal chance 3 Sample size is large np 9 and n 1 p 9 Describes a binomial experiment with normal approximation If p is unknown use worst case 95 confidence Real meaning 95 of samples of this size will produce confidence intervals that capture the true proportion Margin of error Conditions Assumptions independent random 10 of pop 10 success fail Find right population size Central Limit Theory for a Proportion CHAPTER 10 TESTING HYPOTHESES ABOUT PROPORTIONS Stating Hypotheses Null Hypothesis Ho and Alternative Hypothesis HA Two tail test Ho p po and HA p po One tail test Ho p po and HA p po OR Ho p po and HA p po Small p value implies that random variation due to the sampling process alone is not likely to account for the observed difference With small p value we REJECT Ho The true proportion was significantly different from what was stated in Ho Four Steps in Hypothesis Testing 1 Define the hypotheses to test and the required significance level 2 Calculate the value of the test statistic 3 Find the p value based on the observed data 4 State the conclusion Reject the null hypothesis if the p value if p value the data do not provide sufficient evidence to reject the null The significance level is the largest P value tolerated for rejecting a true null hypothesis how much evidence against H0 we require This value is decided arbitrarily before conducting the test The reliability of an interpretation is related to the strength of the evidence The smaller the p value the stronger the evidence against the null hypothesis and the more confident you can be about your interpretation The magnitude or size of an effect relates to the real life relevance of the phenomenon uncovered The p value does NOT assess the relevance of the effect nor its magnitude A confidence interval will assess the magnitude of the effect However magnitude is not necessarily equivalent to how theoretically or practically relevant an effect is ERROR IN HYPOTHESIS TESTS A Type I error is made when we reject the null hypothesis and the null hypothesis is actually true incorrectly reject a true H0 The probability of making a Type I error is the significance level a A Type II error is made when we fail to reject the null hypothesis and the null hypothesis is false incorrectly keep a false H0 The probability of making a Type II error is labeled b The power of a test is 1 CHAPTER 11 CONFIDENCE INTERVALS AND HYPOTHESIS TESTS FOR MEANS Mean x bar Standard deviation n If the population is N then the sample means distribution is N n Large samples are not always attainable Sometimes the cost difficulty or preciousness of what is studied drastically limits any possible sample size Opinion polls have a limited sample size due to time and cost of operation During election times though sample sizes are increased for better accuracy Not all variables are normally distributed o o Income for example is typically strongly skewed Is still a good estimator of m then Central Limit Theorem for Means When randomly sampling from any population with mean m and standard deviation s when n is large enough the sampling distribution of We don t need to take a lot of random samples to rebuild the sampling distribution and find m at its center All we need is one SRS of size n and rely on the properties of the sample means distribution to infer the population mean x is approximately normal N n Confidence interval when is KNOWN When is UNKNOWN For a sample of size n the sample standard deviation s is n 1 is the degrees of freedom The value s n is called the standard error of the mean SEM Scientists often present sample results as mean SEM When s is estimated from the sample standard deviation s the sampling distribution follows a t distribution t m s n with degrees x z n of freedom n 1 When n is very large s is a very good estimate of s and the corresponding t distributions are very close to the normal distribution The t confidence interval close to normal and without outliers When 15 …
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