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# Probs & Stats 2

### probability and statistics concepts

Discrete Finite number
Continuous Infinite without gaps
Nominal Categories No order
Ordinal Ordered but difference between is meaningless Relative comparison Ex:grades
Interval Ordered but meaningful Does not start at 0 Ex:temp
Ratio Ordered/meaningful Starts at 0 Ex:distance
Systematic sample Start point Select every Kth element
Convenience sample Easy to collect Close to researchers location
Stratified sample Subgroups (2 of them) Same characteristics Draw same amt of sample from each Consistent
Cluster sample Divide into groups Select only some groups Choose all elements from selected group Faster and less expensive
Multistage sample Combination of methods Select sample in each stage Each stage different method Natural clusters
Reason for Frequency Tables 1)summarize large data 2)analyze nature 3)basis for graphs
Relative Frequency (Class frequency/sum of all frequency)x 100%
Unusual Center Mean-2s or mean+2s
Coefficient of variation (s/x)*100%
Bimodal 2 modes
Multimodal 2+ modes
Midrange (max # - min#)/2
Percentile L(position)= k(percent)/100 *n
Interquartile Q3-Q1
5 Number Summary(mon,q1,med,q3,max) 1-Vars Stats
Odds against P(not A)/P(A)-> A:B
Payoff odds Net profit: amount of bet
Addition Rule of Probability P(A or B)= P(A) + P(B) /total outcomes
Complements P(none)= 1 - P(at least one) P(at least one) = P(only A or only B or both) = P(A only) + P(B only) - P(both)
Multiplication Rule (Independent) P(A and B) = P(A) * P(B)
Multiplication Rule (Dependent) P(A and B) = P(A) * P(B/A) after event A has occurred
Conditional Probability (Independent) P(B/A) = P(A and B) / P(A)
Bayes' Theorem P(A)*P(B/A) / [P(A)*P(B/A)]+[P(no A)*P(B/no A)]
Fundamental Counting Rule m*n ways P(A) = 1/m*n
Permutation(different) nPr n = # of items r = amt selected
Permutation(identical) nPr = n!/ n1!n2!nk!
Combination Rule nCr no repeats no order
Requirements for Probability Distribution 1)Sum of P(x) = 1 2)0 < P(x) < 1 for every x
Probability mean Sum of x * P(x)
Probability standard deviation Square root of Sum of [(x^2 * P(x0]- mean^2
Unusual Probability P(x or more) < 0.05 P(x or fewer)< 0.05
Binomial Distribution Requirements 1)fixed # of trials 2)independent 3)2 categories 4)P(success)same in all success
Binomial Probability binompdf(n,p(success),x success)
Binomial Mean n*p(success)
Binomial Standard Deviation sq. root of n*P(success)*P(failure)
Unusual values for Binomial mean - 2stan. dev mean + 2stan. dev
Poisson Distribution Requirements 1)x = event of interval 2)random 3)independent 4)uniformly distributed
Poisson Standard Deviaton sq. root of mean
Using Poisson as Binomial Distribution Requirements: 1)n > 100 2) np < 100
Poisson Probability poissonpdf(mean,x selected)
Normal Distribution Characteristics 1)bell-shape 2)mean = 0 3) s = 1
Uniform Distribution Characteristics 1)area = 1 2) correspondence between area and prob
Normal Distribution (Area under graph) normalcdf(left z,right z)
Z-score(normal distribution) invNorm(area left of z-score)
Finding P(individual value) w/ Norm Distr z = x - mean/ stan. dev
Finding P(sample) w/ Norm Distr z = x - mean/ (stan dev./sq. root of n)
mean & x to z-score normalcdf(left, right, mean, s)= P(x) invNorm(P(x))= z-score
Find x value of nonstandard norm distr invNorm(area to left, mean, s)
Sample Variance Sum(x - mean)^2 / n -1 mean = sum of x/n
Normal Distr As Binomial Approximate Requirements 1)independent simple random sample 2)np> 5 and nq>5
Continuity Correction x - 0.5 to x + 0.5
P(Area to left) normalcdf(-99999,z-score)
Normal Distribution on Graph 1)straight line 2)no systematic pattern
Find Critical Value(z-score) 1-(confidence interval/2)-->invNorm(1-alpha)
Margin of Error (z-score) z(alpha/2) * sq. root(p(success)*q(failure)/n) or upper CI - lower CI/ 2
Confidence Interval(z- score) Stat--> 1-PropZInt
P(Success) upper CI + lower CI/ 2
Sample Size(independent) n = (z(alpha/2)^2 *p*q)/E^2 or n = (z(alpha/2)^2 *p*q*.25)/E^2
Margin of Error(stan. dev known) ZInterval Set mean = 0
Sample Size(stan dev known) n= [z(alpha/2)*stan.dev/E]^2
t statistic invT(1-alpha,df=n-1)
Confidence interval(t-score) TInterval
T-Score Properties 1)norm distr 2)different t for different n 3) mean = 0 4)stan. dev. > 1 5)n increase, t --> norm distr
Chi-Square Distribution Properties 1)not symmetric 2)positive values 3)different for each df
Finding Chi-Square 1)calculate alpha and df 2)What kind of test? 3)Look at Tables given
Confidence interval(chi-square) sq.root[(n-1)s^2/chi right] < stan. dev < sq. root[(n-1)s^2/chi left]
Reject null (p-value) p-value < alpha
Find p-value normalcdf(left, right)
Reject null (test statistics) test statistic falls in critical region bounded by critical value
Reject null(confidence interval) Confidence interval does not contain claimed value
Type 1 Error null true --> reject (alpha)
Type 2 Error null false --> fail to reject (beta)
Two Tail Test null = alternate not =
Right Tail Test null = alternate >
Left Tail Test null = alternate <
Testing Claim of Proportion 1-PropZTest
Test Claim on Mean (stan. dev known) Z-Test
Test Claim on Mean (stan dev Unknown) T-Test
Test on 2 Proportions 2-PropZTest
Confidence Interval(2 Proportions) 2- PropZInt
Test on 2 Means (stan. dev. unknown/ independent) 2-SampTTest 2-SampTInt
Test on 2 Means(Stan dev known) 2-SampZTest 2-SampZInt
Claim on Mean (Dependent/differences) 1) L1 - L2 2)TTest 3)TInt
Compare Variation of 2 Samples 2-SampFTest
Correlation 1)Straight-line 2) r = LinRegTTest 3) r > critical value of alpha 4) reject null
Regression LinRegTTest
marginal change slope of regression line
Residual observed y - predicted y y from table - y from regression line
Coefficient of determination r^2
Created by: mnguye2