Both populations have normal distributions 2 the

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Both populations have normal distributions. 2. The variances of the populations are equal ( σ 1 2 = σ 2 2 = σ 2 ). Interval Estimate with σ 2 known where Interval Estimate with σ 2 unknown where Pooled Estimator of the population variance ( σ 2 ): Remarks on Pooled Estimator of σ 2 : 1. A weighted average of the two sample variances. 2.Always between and . 7
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8 Example: Specific Motors “Specific motors” has developed an M-car. 12 “M cars” and 8 “J cars” were tested to compare miles per gallon performance. The sample statistics are Sample #1 Sample #2 M Cars J Cars Sample Size n 1 = 12 cars n 2 = 8 cars Mean = 29.8 mpg = 27.3 mpg Standard Dev. s 1 = 2.56 mpg s 2 = 1.81 mpg Assume that the populations have normal distributions, and have equal variances. Point estimate of the difference between the means: Total degrees of freedom: Pooled Variance: = 95% Confidence Interval: Interpretation: 8
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Hypothesis tests between means of two populations: Hypotheses: H 0 : µ 1 - µ 2 < 0 H 0 : µ 1 - µ 2 > 0 H 0 : µ 1 - µ 2 = 0 H a : µ 1 - µ 2 > 0 H a : µ 1 - µ 2 < 0 H a : µ 1 - µ 2 ≠ 0 In general forms: H 0 : µ 1 - µ 2 < D 0 H 0 : µ 1 - µ 2 > D 0 H 0 : µ 1 - µ 2 = D 0 H a : µ 1 - µ 2 > D 0 H a : µ 1 - µ 2 < D 0 H a : µ 1 - µ 2 D 0 Large Sample (If σ 1 and σ 2 are unknown, s 1 and s 2 are used to estimate them, and z is still valid.) Small Sample Assumptions: 1. Both populations have normal distributions. 2. The variances of the populations are equal ( σ 1 2 = σ 2 2 = σ 2 ). (If σ 1 and σ 2 are known) (If σ 1 and σ 2 are unknown) Rejection Rule: Same as shown in Chapter 9. 9
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10 Example: Par, Inc. (Cont’d p. 4) Sample #1 Sample #2 Par, Inc. Rap, Ltd . Sample Size n 1 = 120 balls n 2 = 80 balls Mean = 235 yards = 218 yards Std Dev. s 1 = 15 yards s 2 = 20 yards (1) Using a 0.01 level of significance, can we conclude that the mean driving distance of Par, Inc. golf balls is greater than the mean driving distance of Rap, Ltd. Golf balls? (2) Using a 0.01 level of significance, can we conclude that the mean driving distance of Par, Inc. golf balls is greater than the mean driving distance of Rap, Ltd. Golf balls by 10 yards ? 10
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Example: Specific Motors Company (Cont’d p. 6) Sample #1 Sample #2 M Cars J Cars Sample Size n 1 = 12 cars n 2 = 8 cars Mean = 29.8 mpg = 27.3 mpg Standard Dev. s 1 = 2.56 mpg s 2 = 1.81 mpg (1) Can we conclude at a 0.05 level of significance that the miles per gallon performance of M-cars is greater than the miles per gallon performance of J cars?
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