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**Chapter 9**Hypothesis Testing and Estimation for Two Population Parameters**Chapter 9 - Chapter Outcomes**After studying the material in this chapter, you should be able to: Use sample data to test hypotheses that two population variances are equal. Discuss the logic behind, and demonstrate the techniques for, using sample data to test hypotheses and develop interval estimates about the difference between two population means for both independent and paired samples.**Chapter 9 - Chapter Outcomes(continued)**After studying the material in this chapter, you should be able to: Carry out hypotheses tests and establish interval estimates, using sample data, for the difference between two population proportions.**Hypothesis Tests for Two Population Variances**HYPOTHESIS TESTING STEPS • Formulate the null and alternative hypotheses in terms of the population parameter of interest. • Determine the level of significance. • Determine the critical value of the test statistic. • Select the sample and compute the test statistic. • Compare the calculated test statistic to the critical value and reach a conclusion.**Hypothesis Tests for Two Population Variances**Two-Tailed Test Upper One-Tailed Test Lower One-Tailed Test Format 1**Hypothesis Tests for Two Population Variances**Two-Tailed Test Upper One-Tailed Test Lower One-Tailed Test Format 2**Hypothesis Tests for Two Population Variances**Two-Tailed Test Upper One-Tailed Test Lower One-Tailed Test Format 3**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 Rejection Region /2 = 0.05 F 0 Since F=1.47 F= 2.75, do not reject H0**Independent Samples**Independent samples are those samples selected from two or more populations in such a way that the occurrence of values in one sample have no influence on the probability of the occurrence of values in the other sample(s).**Hypothesis Tests for Two Population Means**Two-Tailed Test Upper One-Tailed Test Lower One-Tailed Test Format 1**Hypothesis Tests for Two Population Means**Two-Tailed Test Upper One-Tailed Test Lower One-Tailed Test Format 2**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 9-3)**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 AND12 = 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)**- $330.45 $1,458.33**Confidence Interval Estimates for 1 - 2**STANDARD DEVIATIONS UNKNOWN AND12 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 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 4.928 9.272**Hypothesis Tests for Two Population Proportions**Two-Tailed Test Upper One-Tailed Test Lower One-Tailed Test Format 1**Hypothesis Tests for Two Population Proportions**Two-Tailed Test Upper One-Tailed Test Lower One-Tailed Test Format 2**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**Independent Samples**Paired Samples Key Terms