Ch13 - Chapter 13 Inference about Two Populations 1 12.1...

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1 Inference about Two Populations Chapter 13
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2 12.1 Introduction Variety of techniques are presented whose objective is to compare two populations. We are interested in: The difference between two means. The ratio of two variances. The difference between two proportions.
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3 Two random samples are drawn from the two populations of interest. Because we compare two population means, we use the statistic . 13.2 Inference about the Difference between Two Means: Independent Samples 2 1 x x -
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4 1. is normally distributed if the (original) population distributions are normal . 2. is approximately normally distributed if the (original) population is not normal, but the samples’ size is sufficiently large (greater than 30). 3. The expected value of is μ 1 - μ 2 4. The variance of is σ 1 2 / n 1 + σ 2 2 / n 2 2 1 x x - 2 1 x x - The Sampling Distribution of 2 1 x x - 2 1 x x - 2 1 x x -
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5 If the sampling distribution of is normal or approximately normal we can write: Z can be used to build a test statistic or a confidence interval for μ 1 - μ 2 2 1 2 1 n n ) ( ) x x ( Z 2 2 2 1 2 1 σ + σ μ - μ - - = 2 1 x x - Making an inference about μ 1 μ 2
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6 2 1 2 1 n n ) ( ) x x ( Z 2 2 2 1 2 1 σ + σ μ - μ - - = Practically, the “Z” statistic is hardly used, because the population variances are not known. ? ? Instead, we construct a t statistic using the sample “variances” (S 1 2 and S 2 2 ). S 2 2 S 1 2 t Making an inference about μ 1 μ 2
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7 Two cases are considered when producing the t-statistic. The two unknown population variances are equal . The two unknown population variances are not equal . Making an inference about μ 1 μ 2
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8 Inference about Inference about μ μ 1 1 μ μ 2 2 : Equal : Equal variances variances 2 n n s ) 1 n ( s ) 1 n ( S 2 1 2 2 2 2 1 1 2 p - + - + - = Example: s 1 2 = 25; s 2 2 = 30; n 1 = 10; n 2 = 15. Then, 04347 . 28 2 15 10 ) 30 )( 1 15 ( ) 25 )( 1 10 ( S 2 p = - + - + - = Calculate the pooled variance estimate by: n 2 = 15 n 1 = 10 2 1 S 2 2 S The pooled variance estimator
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9 Inference about Inference about μ μ 1 1 μ μ 2 2 : Equal : Equal variances variances 2 n n s ) 1 n ( s ) 1 n ( S 2 1 2 2 2 2 1 1 2 p - + - + - = Example: s 1 2 = 25; s 2 2 = 30; n 1 = 10; n 2 = 15. Then, 04347 . 28 2 15 10 ) 30 )( 1 15 ( ) 25 )( 1 10 ( S 2 p = - + - + - = Calculate the pooled variance estimate by: 2 p S n 2 = 15 n 1 = 10 2 1 S 2 2 S The pooled Variance estimator
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10 Inference about Inference about μ μ 1 1 μ μ 2 2 : Equal : Equal variances variances Construct the t-statistic as follows: 2 n n . f . d ) n 1 n 1 ( s ) ( ) x x ( t 2 1 2 1 2 p 2 1 - + = + μ - μ - - = 2 1 Perform a hypothesis test H 0 : μ 1 2 = 0 H 1 : μ 1 2 > 0 or < 0 or 0 Build a confidence interval level. confidence the is where ) n 1 n 1 ( s t ) x x ( 2 1 2 p 2 1 α - 1 + ± - 2 α
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11 1 ) ( 1 ) ( ) / ( d.f. ) ( ) ( ) ( 2 2 2 2 2 1 2 1 2 1 2 2 2 2 1 2 1 2 2 2 1 2 1 2 1 - + - + = + - - - = 2 1 n n s n n s n s n s n s n s x x t μ Inference about μ 1 μ 2 : Unequal variances
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12 Inference about μ 1 μ 2 : Unequal variances Conduct a hypothesis test as needed, or, build a confidence interval level confidence the is where n s n s 2 t x x interval Confidence - 1 + ± - α ) 2 2 2 1 2 1 ( ) 2 1 (
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13 Which case to use: Equal variance or unequal variance?
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This note was uploaded on 02/07/2012 for the course BUS 265 taught by Professor Kim during the Summer '11 term at Minnesota.

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Ch13 - Chapter 13 Inference about Two Populations 1 12.1...

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