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General ‘Guarvnatre Cater ‘Onesample teat for mean One sample tt for proportion anviate, Qusettatve eat frida contin yp te Population eons, fer each uso the cateporical arabe | MSAmOe est for rans Guan itatne Categorical ‘More than two soups? Sapulaton proportions pf fOr ach group ofthe orpiansiony egerical vara ANOVA F Ret for eas two-sample zt for proportion aap eangoreat Aelatonahp betwee vrs ina fe eerie esas at tn iy tes Mean # £ Variance a 2 Propertion P * Cconrtation ’ r Inference and Comparison of Means == NOH) tra Z-statistie: Used when o is known Different than z-score: Z-statistic represents | sample in population rather than | value in sample Same applies for 2 populations, z-statistic represents 2 samples from 2 populations (1 from each) rather than 2 values from 2 samples ‘T-statistie: Used when o is unknown, replaced with sample standard de Standard error: s/sqrt(n) estimates standard error, the sample standard devi cannot be known if p is unknown. Follows follows t-distribution with n - | degrees of freedom (df) ‘T-distribution: A symmetric, unimodal density curve with larger spread than the standard normal density curve Specified through df (represents number of observations used to estimate c using s) ‘T-distributions with smaller df have greater spread and larger df As n increase: the t(n - 1) distribution gets close to the N(0, 1) distribution tdistribution vs. N(O, 1) ‘ont eane tall sped Density carve symmetric (ensly cere ceded a= trio Standard easton aft darbuton > {aritonspected by Narmal eechd oy wand Sample size for quantitative variables: n< 13: Only use t if data generally symmetric, unimodal, no outliers n=> 15; Proceed in absence of extreme skewness or large outliers n> 40: Proceed 1 sample t-test: Cumulative proportion to find p-value: pi(t, df) Example: Ho: = 21, Ha: pw >21, n= 101, t= 4.55 — p-value = P((100) > 4.55) = pt(4.55, 100) = 0.000007 (greater or less sign aligns with Ha) 1 sample t-confidence interval for uz Bends: . ~ Rie tis the critical value from the t(n - 1) densi t* chosen so that C% of values in the distribution are between -t* and t* Inverse calculation (percentiles) to find t*: qt (percentile, dé) Example: C = 95%, n= 31 — t* = qt(0.025, 30) Sampling distributions for comparing 2 means: + Appving ou wes for combining random varabiss —[2,-¥ (ut) Mon of =e sone + vatanoe of = 42 8a, + 7a, assuming ncepencece + andar evton ot, — fy i8 a, + Yn sams 1 (san E+) 2 sample t-test: Simple way to find df: Choose smaller of nl - | and n2 -L Conservative confidence and significance levels Example: Ho: yl = 2, Ha: al > 2, nl =21,n2 = 23, t= 2.31 > p-value: P(t(20) => 2.31)= 1 = pt(2.31, 20)= 0.016 Equal variances between 2 populations: + Ihwe betieve that parameter a? gf, then we anly have one variance =) lasmna stent 6 = t(ay +m: -2) Seater ests = HaN=twam) veo.1) “ft * Estrnatng the pooled wartance x2: Yestns ' 2 sample t-confidence interv: Solve for t* using same inverse calculation method as | sample #2 t-eonfidence interval with appropriate (smaller) df aon te [2 1 Inference and Comparison of Proportion: Sample proportion: Phat = X (successes in sample)/n Estimation for population proportion Can be defined using xbar Sampling distribution of sample proportion: #402) - -r(> =) Only parameter, p, needed to approximate sampling distribution since both mean and standard deviation can be calculated from it Standard deviation maximized when p = 0.5 Sample size minimum for categorical variables: np => 10, n(1 - p) Significance test for p: Ho: P= Be HB > Py ORM:
