Probablity the proportion of expected outcomes vs actual outcomes Randomness individual outcomes are uncertain but there is a regular distribution of outcomes in a large number of reps Probability Model 1 Sample Space S a set or list of all possible outcomes of a random process An event is a subset of the sample space 2 A probability for each possible event in the sample space is P A Size of Event Total of Outcomes Simple Events Contain only one outcome Are mutually exclusive P A 0 A is an impossible event 0 P A 1 Rules to Know mutually exclusive Addition Rule Or Rule P A or B P A P B if and only if A and B are Complement Rule All outcomes not in A P AC 1 P A General Addition for any events A B P A or B P A P B P A and B Multiplication Rule P A and B P A P B if A and B are independent Independence two events are independent if the outcomes of one doesn t influence to occurrence of the other ex Flipping a coin probability is always 5 for H T Mutually exclusive events are NOT independent Dependent Events the occurrence of one influences the occurrences of another Conditional Probability Random Variable a numerical measurement of the outcome of a random process Discrete Random Variable a variable that can assume only a finite value Continuous Random Variable assume any value on an interval decimals included Probability Distribution Function PDF X P X X1 X2 P1 P2 X3 P3 Xk Pk Expected Value E x x1p1 x2p2 xkpk E x np Variance 2 a parameter measuring how far a set of numbers is spread out usually how far numbers lie from the mean The larger the variance the more scatter there tends to be Var x E x 2 or Var X npq where q probability of failure Var x x1 2p1 x2 2p2 xk 2pk Empirical Rule 68 of all points between and 95 of all points are between 2 and 99 7 of all points are between 3 and 3 2 Confidence Interval CI gets narrower when we decrease confidence level CI gets wider when we increase confidence level 95 confidence means that 95 samples of this size will yield CI s that capture the true proportion p z p 1 p n Hypotheses H0 null hypothesis effect is due to chance p po Ha alternative hypothesis effect is due to something else trying to prove o p po ONE TAIL o p po ONE TAIL o p po TWO TAIL P Values the probability of seeing something as extreme or even more extreme than what was already observed o A small p value is evidence against initial assumption thus evidence towards Ha o Always say Fail to reject Ho if no proof is found towards Ha o P values less than 0 05 are usually considered significant o If P then we reject Ho o If P then we fail to reject H0 Type I Error when the null hypothesis is rejected but it is actually true Type II Error when we fail to reject the null hypothesis and the null hypothesis is actually false Factorials n k n k k n Binomial Distribution Formula n n n 1 n 2 n k 1 n k o P x n x n x px qn x USEFUL CRITICAL VALUES Normal Distribution o Mean Median Mode o Location determined by mean spread determined by standard deviation Finding Sample Size for Margin of Error ME CI 0 90 0 95 0 98 0 99 z score 1 645 1 96 2 326 2 576 p p0 z p0 1 p0 n z 2 p 1 p ME 2 n Z Score Formula z x Test Statistic Formula Continuity Correction z x probability of failure np npq where n population p probability of success q
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