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College Prep Stats Chapter 9 Important Info Sheet Section 9.2 Notation: For population 1, we let: For population 2, we let: p1 = population proportion n1 = size of the sample x1 = number of successes in the sample p2 = population proportion n2 = size of the sample x2 = number of successes in the sample x1 (the sample proportion) n1 qˆ1 1 pˆ1 x2 (the sample proportion) n2 qˆ2 1 pˆ 2 pˆ1 pˆ 2 For Null and Alternative Hypotheses: H 0 : p1 p2 H1 : p1 p2 , H1 : p1 p2 , H1 : p1 p2 Pooled estimate x x p 1 2 n1 n2 Test Statistic for Two Proportions ( pˆ1 pˆ 2 ) , (ZPROP Program) z pq pq n1 n2 Critical values: *left tailed test, α is in the left tail, z* = invNorm(area in the left tail, 0, 1) *right tailed test, α is in the right tail, z* = invNorm(area in the left tail, 0, 1) 𝛼 *two tailed test, α is divided equally between the two tails, z* = invNorm( , 0, 1) 2 P-values: For right-tailed tests: P(z > test statistic) *To find this probability in your calculator, type: normalcdf(z test statistic, 99999, 0, 1) For left-tailed tests: P(z < –test statistic) *To find this probability in your calculator, type: normalcdf(–99999, –z test statistic, 0, 1) ***Don’t forget if your test is two-sided, double your P-value*** Confidence Interval Estimate of 𝒑𝟏 − 𝒑𝟐 ( pˆ1 pˆ 2 ) E, ( pˆ1 pˆ 2 ) E pˆ qˆ pˆ qˆ where E | z* | 1 1 2 2 , (EPROP program) n1 n2 MAKE SURE THAT z* IS POSITIVE WHEN YOU PLUG IT INTO YOUR CALCULATOR!!!!!!! **Your Margin of Error should never be negative** Section 9.3 Independent Samples with σ1 and σ2 Unknown and Not Assumed Equal Notation For population 1, we let: For population 2, we let: 1 = population mean 1 = population standard deviation 2 = population mean 2 = population standard deviation n1 = size of the first sample x1 = sample mean s1 = sample standard deviation n2 = size of the first sample x2 = sample mean s2 = sample standard deviation For Null and Alternative Hypotheses: H 0 : 1 2 H1 : 1 2 , H1 : 1 2 , H1 : 1 2 Test Statistic for Two Means: Independent Samples t ( x1 x2 ) s12 n1 s2 2 n2 , (TMEAN program) Degrees of freedom: df = n – 1 of the smaller sample Critical values: Degrees of freedom, df = n – 1 of the smaller sample *left tailed test, α is in the left tail, t* = invT(area in the left tail, df) *right tailed test, α is in the right tail, t* = invT(1 – area in the right tail, df) 𝛼 *two tailed test, α is divided equally between the two tails, t* = invT( , df) 2 P-values: Degrees of freedom, df = n – 1 of the smaller sample For right-tailed tests: P(t > test statistic) *To find this probability in your calculator, type: tcdf(t test statistic, 99999, df) For left-tailed tests: P(t < –test statistic) *To find this probability in your calculator, type: tcdf(–99999, t test statistic , df) ***Don’t forget if your test is two-sided, double your P-value*** Confidence Interval Estimate of 1 2 : Independent Samples ( x1 x2 ) E, ( x1 x2 ) E s2 where E | t* | 1 n1 s2 2 , (EMEAN Program) n2 MAKE SURE THAT t* IS POSITIVE WHEN YOU PLUG IT INTO YOUR CALCULATOR!!!!!!! **Your Margin of Error should never be negative** Section 9.4 Notation for Dependent Samples d = individual difference between the two values of a single matched pair µ = mean value of the differences d for the population of paired data d d = mean value of the differences d for the paired sample data (equal to the mean of the x – y values) sd = standard deviation of the differences d for the paired sample data n = number of pairs of data. For Null and Alternative Hypotheses: H 0 : d 0 H1 : d 0, H1 : d 0, H1 : d 0 Hypothesis Test Statistic for Matched Pairs t d d , where degrees of freedom = n – 1 sd n Critical Values: Degrees of freedom (df) = n – 1. *left tailed test, α is in the left tail, t* = invT(area in the left tail, df) *right tailed test, α is in the right tail, t* = invT(1 – area in the right tail, df) 𝛼 *two tailed test, α is divided equally between the two tails, t* = invT( , df) 2 P-values: Degrees of freedom (df) = n – 1. For right-tailed tests: P(t > test statistic) *To find this probability in your calculator, type: tcdf(t test statistic, 99999, df) For left-tailed tests: P(t < –test statistic) *To find this probability in your calculator, type: tcdf(–99999, t test statistic , df) ***Don’t forget if your test is two-sided, double your P-value*** Confidence Intervals for Matched Pairs d E, d E where E | t* | sd n MAKE SURE THAT t* IS POSITIVE WHEN YOU PLUG IT INTO YOUR CALCULATOR!!!!!!! **Your Margin of Error should never be negative**