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Type I Error (a): Incorrectly reject Ho when it is true. ; Inpo IF Emner LB): Fail to reject Ho when He is true. Hypothesis Testing for Two Means : 1) Null Hypothesis ( Ho): Assumes no difference between two population means (Ho:14=ne). Alternative Hypothesis : Assumes a difference exists ( HI: mitz, MIDNZ , or Mismz). 2 ) Independent Samples t-test : Used when comparing means from two different groups, both samples assumed to be independent. Paired Samples to test : Used when comparing means from the same group at different times leg. before and after treatment). 3 ) The data is normally distributed ( important for small sample sizes ), and the two populations' variances are equal for independent samples t-tot. 4 ) test statistic for independent samples : = spots , where sp = (-Do +las- 1hit If = 1+ 1 2 - 2 test statistic for paired samples : t = son SOAR , Di=Xi-YU, D= ELL, So = 2401-D, If=N-I 5 ) If on 2 and oz are known , andlor of N1 7130 and 1271 30 , test statistic for independent samples: Z = 4- 12 If n 7130 and population variance of difference Op is known, test statistic for paired samples : Z= 6 ) Using significance level a and If, find critical value. Or, use test statistic to find p-value. Depending on whether Hi is two- tuiled or one-tailed , make decision about Ho 1 . . Hypothesis Testing for One Proportion : 1) Sample Proportion (P): The proportion of successes in a sample, from p = mix us successes. L. ( 2 ) Null Hypothesis ( Ho) : Assume P equal to specific value ( Ho : p=po). Alternative Hypothesis: What you want to prove ( Hi: pape, pape, papa), 1 (3) Sample should be random, and sample size should be large enough lapo 725 and A ( 1-pol 7:5 ) to use normal approximation. [ 4 ) test statistic : Z= ; Find p-value from test statistic and reject Ho of p-values . Hypothesis Testing for Two Proportions: 1 ) Sample Proportions: Pi = M , pz = As , with 21 and zz as maker of successes. - (2 ) Null Hypothesis: The turo population proportions are equal.(to: pi=p2). Alternative Hypothesis: The twoaren't qual (.HI: p1#p2, plzpz, pisp2). Pi- where P = $1$22 [* 3) test statistic : Z= Spy-PJ() 21712 . Use the pooled proportion ( P) when Ho: p1=p2. If Ha: pitpz , pispz, orpizpz , test statistic : Z = pups [ : 4) Assume samples are independent, and sample size is large enough. (wipi 215, will-pil 215 , 1273205, As (653) 715) for normal approx. (5 ) Find critical value using a and Z-table . Or find p- value from Z test statistic. If the absolute value of Z is greater than the critical value, or if p-value so , reject Ho . CI for Variances : 1 ) Formula : ( 2 2/2 ( or 1) " Hypothec's Testing for One Variance : 1) Ho: Variance is equal to specified . ?. HI: Variance is not equal to os " ( two-tailed ) or is "( greater / less than 08 2 (one-tailed ) . 2 ) Test Statistic : 72 = fallsz 0oz , withs as sample variance and 0, 2 as mill variance . [ 2 3 ) Find critical valuels ) from X " bable based on significance level . and dif =n- 1, Or find p-value from test statistic. If What ( 7 calator What ? 12, 2 (nell, reject for two-sided. For Xp lil) = X test, p-value= Bloboften call. [ One - Sided CIs: 1 )One-sided Upper CI for Mean! I + Zac. (); One-sided Lower CI, for Mom :5-Zac (JE) ( 2 ) One -sided Upper CI for Proportions: P + Zac ELECT ; One-sided Lower CI for Mean: P - Zx PAPI. 3) One-sided Upper CI for Variances: feller TheWIT ; One-sided Lower CI for Variances : th lol) | Abs . Value Z probs . : _ P ( 1 7 /