notes Ch 10 - CHAPTER 10 Comparisons Involving Means 10.1...

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CHAPTER 10 Comparisons Involving Means 10.1 Difference Between Two Population Means: σ 1 and σ 2 Known Interval Estimation of μ 1 and μ 2 Let μ 1 = mean of population 1 μ 1 = mean of population 2 1 x = sample mean of population 1 2 x = sample mean of population 2 n 1 = sample size of sample from population 1 n 2 = sample size of sample from population 2 σ 1 = standard deviation of population 1 σ 2 = standard deviation of population 2 Point estimator of the difference between two population means: 2 1 x x - Assuming both samples are independent simple random samples and both populations are normally distributed, the interval estimate of the difference between two population means when σ 1 and σ 2 are known is: 2 2 2 1 2 1 2 / 2 1 n n z x x σ σ α + ± - Where 2 2 2 1 2 1 2 / n n z σ σ α + is the margin of error 1 – α is the confidence coefficient 2 / α z is the z-score at α/2. Z can be obtained using Normal distribution tables or in Excel with the formula =NORMSINV(p), where p = 1- α/2. Example: Greystone Department Stores operates two stores: one in the inner city and the other in a suburban shopping center. The store collected demographic data including age from samples of customers at both locations. The results are provided below. At 95% confidence, is the mean population age statistically different between these two locations? Find the point estimate and the interval estimate of the difference between these two populations. Inner-City Store Suburban Store Sample Size n 1 = 36 n 2 = 49 Sample Mean 1 x = 40 years 2 x = 35 years Population Standard Deviation σ 1 = 9 years σ 2 = 10 years 1
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95% confidence corresponds to α = .05 and z α/2 = NORMSINV(1-α/2) = NORMSINV(1-.05/2) = 1.96. The point estimate of the difference between costumers’ age of Inner-City store and Suburban store is: 2 1 x x - = 40 – 35 = 5 years The interval estimate of the difference between these two populations is: 2 2 2 1 2 1 2 / 2 1 n n z x x σ σ α + ± - 49 10 36 9 96 . 1 35 40 2 2 + ± - 06 . 4 5 ± The margin of error is 4.06. The interval estimate of the difference between the two populations is 5 – 4.06 = .94 years to 5 + 4.06 = 9.06 years. Most used Z-scores 1- α α α/2 Z .95 .05 .025 1.96 .99 .01 .005 2.576 .90 .10 .05 1.645 Hypothesis Tests About μ 1 – μ 2 Let H o is the null hypothesis H a is the alternative hypothesis D o is the hypothesized difference between μ 1 and μ 2 The three possible scenarios are: Scenario 1 2 3 Hypotheses H o : μ 1 – μ 2 ≥ D o H a : μ 1 – μ 2 < D o H o : μ 1 – μ 2 ≤ D o H a : μ 1 – μ 2 > D o H o : μ 1 – μ 2 = D o H a : μ 1 – μ 2 ≠ D o Reject H o if p-value ≤ α p-value ≤ α p-value ≤ α/2 The test statistic for hypothesis tests about μ 1 – μ 2 when σ 1 and σ 2 are known is: 2
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2 2 2 1 2 1 2 1 ) ( n n D x x z o σ σ + - - = Example: As part of a study to evaluate differences in education quality between two training centers, a standardized examination is given to individuals who are trained at the centers. The difference between the mean examination scores is used to assess quality differences between the centers. Based on the data given below, do these data suggest a
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