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ch_10_Notes

# ch_10_Notes - Chapter 10 Statistical Inference about Means...

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Chapter 10 Statistical Inference about Means and Proportions with Two Populations Outline: Estimation of the difference between means of two populations: 1. Point estimator of the difference between the means of the populations. 2. Sampling distribution of the point estimator. 3. Interval estimate of μ 1 2 : Large Sample Case. 4. Interval estimate of μ 1 2 : Small Sample Case. Hypothesis tests about the difference between the means of two populations: Independent Samples. Inferences about the difference of means: Matched Samples. Point Estimator for μ 1 2 1. Select a simple random sample of size n 1 from population 1 and compute 1 x . 2. Select a simple random sample of size n 2 from population 2 and compute 2 x . 3. Compute the point estimator: 2 1 x x - . 1

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Sampling Distribution of 2 1 X X - : Expected Value of 2 1 X X - : The point estimator of the difference of means ( 2 1 μ μ - ) is 2 1 X X - Standard Deviation of 2 1 X X - : (Aside) An easier way to remember is that where σ 1 = standard deviation of population 1 σ 2 = standard deviation of population 2 n 1 = sample size from population 1 n 2 = sample size from population 2 2 2 1 2 1 ) ( μ - μ = - X X E 2 2 2 1 2 1 2 1 n n X X σ + σ = σ - 2 2 2 2 1 2 1 X X X X σ + σ = σ -
Interval Estimate of μ 1 2 : Large Sample (n 1 > 30 and n 2 > 30) σ 1 known and σ 2 known where 1- α is the confidence coefficient σ 1 unknown and σ 2 unknown where 3 2 1 2 / 2 1 ) ( X X z x x - α σ ± - 2 1 2 / 2 1 ) ( X X s z x x - α ± - 2 2 2 1 2 1 2 1 n s n s s X X + = -

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Example: Par, Inc. Par, Inc. is a manufacturer of golf equipment and has developed a new golf ball to provide extra distance . In a test of the driving distance using a mechanical device, a sample of golf balls was compared with a sample of golf balls from a competitor (Rap, Ltd.) Sample Statistics Sample #1 Sample #2 Par, Inc. Rap, Ltd . Sample Size n 1 = 120 balls n 2 = 80 balls Mean 1 x = 235 yards 2 x = 218 yards Standard Dev. s 1 = 15 yards s 2 = 20 yards Point Estimate of the difference between the populations, where μ 1 is the mean driving distance of golf balls from Par, Inc.; μ 2 is the mean driving distance of golf balls from Rap, Ltd. Point Estimate of Point Estimate of standard deviation of is given by 95% confidence interval: 4 2 1 μ μ - 2 1 X X - 2 1 X X s -
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Interval Estimate of μ 1 2 : Small Sample (n 1 < 30 and/or n 2 < 30) Assumptions: 1. Both populations have normal distributions.
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