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 Format 1 Two-Tailed Test Upper One- Tailed Test Lower One- Tailed Test

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

Hypothesis Tests for Two Population Variances Format 3 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: n i = Sample size from ith population n j = Sample size from jth population s i 2 = Sample variance from ith population s j 2 = Sample variance from jth population

Hypothesis Tests for Two Population Variances (Example 9-2) F 0 df: D i = 10, D j =12 Rejection Region  /2 = 0.05 Since F=1.47  F= 2.75, do not reject H 0

Independent Samples 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 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: s 1 2 = Sample variance from population 1 s 2 2 = Sample variance from population 2 n 1 and n 2 = Sample sizes from populations 1 and 2 respectively

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

Hypothesis Tests for Two Population Means (Example 9-3) Rejection Region  /2 = Since t < 2.048, do not reject H 0 Rejection Region  /2 = 0.025

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  1 2 =  2 2 where: = Pooled standard deviation t  /2 = critical value from t-distribution for desired confidence level and degrees of freedom equal to n 1 + n 2 -2

Confidence Interval Estimates for  1 -  2 (Example 9-5) - $ $1,458.33

Confidence Interval Estimates for  1 -  2 STANDARD DEVIATIONS UNKNOWN AND  1 2   2 2 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 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 x 1 and x 2 = Values from sample 1 and 2, respectively

Paired Samples Hypothesis Testing and Estimation MEAN PAIRED DIFFERENCE where: d i = i th paired difference n = Number of paired differences

Paired Samples Hypothesis Testing and Estimation STANDARD DEVIATION FOR PAIRED DIFFERENCES where: d i = i th 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 s d = 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= < 1.833, do not reject H 0

Paired Samples Hypothesis Testing and Estimation PAIRED CONFIDENCE INTERVAL ESTIMATE

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

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: x 1 and x 2 = 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 p 1 and p 2 = 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 < , reject H 0

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

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

Key Terms Independent Samples Paired Samples