Two Sample Tests Ho Ho Ha Ha TEST FOR EQUAL VARIANCES TEST FOR EQUAL MEANS Ho Ho Population 1 Population 1 Population 2 Population 2 Ha Ha Population 1 Population 2 Population 1 Population 2

Hypothesis Tests for Two Population Variances Two-Tailed Test Upper One-Tailed Test Lower One-Tailed Test

Hypothesis Tests for Two Population Variances F-TEST STATISTIC FOR TESTING WHETHER TWO POPULATIONS HAVE EQUAL VARIANCES where: ni = Sample size from ith population nj = Sample size from jth population si2= Sample variance from ith population sj2= Sample variance from jth population

Hypothesis Tests for Two Population Variances (Example 9-2) df: Di = 10, Dj =12 a = .10 Rejection Region /2 = 0.05 F = 1.47 F Since F=1.47  F/2= 2.76, do not reject H0

Independent Samples Independent samples Selected from two or more populations Values in one sample have no influence on the values in the other sample(s).

Hypothesis Tests for Two Population Means Format 1 Two-Tailed Test Upper One-Tailed Test Lower One-Tailed Test

Hypothesis Tests for Two Population Means Format 2 Two-Tailed Test Upper One-Tailed Test Lower One-Tailed Test

Hypothesis Tests for Two Population Means T-TEST STATISTIC (EQUAL POPULATION VARIANCES) where: Sample means from populations 1 and 2 Hypothesized difference Sample sizes from the two populations Pooled standard deviation

Hypothesis Tests for Two Population Means POOLED STANDARD DEVIATION Where: s12 = Sample variance from population 1 s22 = Sample variance from population 2 n1 and n2 = Sample sizes from populations 1 and 2 respectively

Hypothesis Tests for Two Population Means (Unequal Variances) t-TEST STATISTIC where: s12 = Sample variance from population 1 s22 = Sample variance from population 2

Hypothesis Tests for Two Population Means (Example 9-4) Rejection Region  /2 = 0.025 Rejection Region  /2 = 0.025 Since t < 2.048, do not reject H0

Hypothesis Tests for Two Population Means DEGREES OF FREEDOM FOR t-TEST STATISTIC WITH UNEQUAL POPULATION VARIANCES

Confidence Interval Estimates for 1 - 2 STANDARD DEVIATIONS UNKNOWN AND 12 = 22 where: = Pooled standard deviation t/2 = critical value from t-distribution for desired confidence level and degrees of freedom equal to n1 + n2 -2

Confidence Interval Estimates for 1 - 2 (Example 9-5) - $330.46 $1,458.34

Confidence Interval Estimates for 1 - 2 STANDARD DEVIATIONS UNKNOWN AND 12  22 where: t/2 = critical value from t-distribution for desired confidence level and degrees of freedom equal to:

Confidence Interval Estimates for 1 - 2 LARGE SAMPLE SIZES where: z/2 = critical value from the standard normal distribution for desired confidence level

Paired Samples Hypothesis Testing and Estimation Paired samples are samples that selected such that each data value from one sample is related (or matched) with a corresponding data value from the second sample. The sample values from one population have the potential to influence the probability that values will be selected from the second population.

Paired Samples Hypothesis Testing and Estimation PAIRED DIFFERENCE where: d = Paired difference x1 and x2 = Values from sample 1 and 2, respectively

Paired Samples Hypothesis Testing and Estimation MEAN PAIRED DIFFERENCE where: di = ith paired difference n = Number of paired differences

Paired Samples Hypothesis Testing and Estimation STANDARD DEVIATION FOR PAIRED DIFFERENCES where: di = ith paired difference = Mean paired difference

Paired Samples Hypothesis Testing and Estimation t-TEST STATISTIC FOR PAIRED DIFFERENCES where: = Mean paired difference d = Hypothesized paired difference sd = Sample standard deviation of paired differences n = Number of paired differences

Paired Samples Hypothesis Testing and Estimation (Example 9-6) Rejection Region  = 0.05 Since t=0.9165 < 1.833, do not reject H0

Paired Samples Hypothesis Testing and Estimation PAIRED CONFIDENCE INTERVAL ESTIMATE

Paired Samples Hypothesis Testing and Estimation (Example 9-7) 95% Confidence Interval 4.927 9.273

Hypothesis Tests for Two Population Proportions Format 1 Two-Tailed Test Upper One-Tailed Test Lower One-Tailed Test

Hypothesis Tests for Two Population Proportions Format 2 Two-Tailed Test Upper One-Tailed Test Lower One-Tailed Test

Hypothesis Tests for Two Population Proportions POOLED ESTIMATOR FOR OVERALL PROPORTION where: x1 and x2 = number from samples 1 and 2 with desired characteristic.

Hypothesis Tests for Two Population Proportions TEST STATISTIC FOR DIFFERENCE IN POPULATION PROPORTIONS where: (1 - 2) = Hypothesized difference in proportions from populations 1 and 2, respectively p1 and p2 = Sample proportions for samples selected from population 1 and 2 = Pooled estimator for the overall proportion for both populations combined

Hypothesis Tests for Two Population Proportions (Example 9-8) Rejection Region  = 0.05 Since z =-2.04 < -1.645, reject H0

Confidence Intervals for Two Population Proportions CONFIDENCE INTERVAL ESTIMATE FOR 1- 2 where: p1 = Sample proportion from population 1 p2 = Sample proportion from population 2 z = Critical value from the standard normal table

Confidence Intervals for Two Population Proportions (Example 9-10) -0.034 0.104

Key Terms Independent Samples Paired Samples