Chapter Nine

Chapter Nine - Lecture Notes Chapter Nine: Inferences Based...

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Lecture Notes Chapter Nine: Inferences Based on Two Samples: Confidence Intervals and Tests of Hypothesis Randall Miller 1 | Page 1. Identifying the Target Parameter Determining the Target Parameter Parameter Keys Words or Phrases Type of Data 12 µµ Mean difference; difference in averages Quantitative pp Difference between proportions, percentages, fractions, or rates; compare proportions Qualitative 22 / σσ Ratio of variances; difference in variability or spread; compare variation Quantitative 2. Comparing Two Population Means: Independent Sampling Properties of the Sampling Distribution of ( ) x -x 1. The mean of the sampling distribution of ( ) xx is ( ) . 2. If the two samples are independent, the standard deviation of the sampling distribution is ( ) nn σ = Where 2 1 and 2 2 are the variances of the two populations being sampled 1 n and 2 n and are the respective sample sizes. We also refer to ( ) as the standard error of the statistic ( ) . 3. By the central limit theorem, the sampling distribution of ( ) is approximately normal for large samples . Large Sample Confidence Interval for ( ) μ -μ ( ) ( ) ( ) ( ) 1 2 /2 1 2 1 2 xx z ss αα α −± =−± + ≈−± +
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Lecture Notes Chapter Nine: Inferences Based on Two Samples: Confidence Intervals and Tests of Hypothesis Randall Miller 2 | Page Large-Sample Test of Hypothesis for ( ) 12 μ -μ One-Tailed Test [ ] 01 2 0 0 0 : : or : a a HD µµ −= −< −> Two-Tailed Test 0 0 : : a −≠ Where 0 D = Hypothesized difference between the means (this difference is often hypothesized to be equal to 0) Test statistic : ( ) ( ) 0 xx xx D z σ −− = where ( ) 2 2 22 1 2 1 2 ss n n nn σσ = +≈ + Rejection region : zz α <− [or > when 0 : a ] Rejection region : /2 > Conditions Required for Valid Large-Sample Inferences about ( ) 1. The two samples are randomly selected in an independent manner from the two target populations. 2. The sample sizes 1 n and 2 n , both large (i.e., 30 and 30 ≥≥ ). (By the central limit theorem, this condition guarantees that the sampling distribution of ( ) will be approximately normal, regardless of the shapes of the underlying probability distributions of the populations. Also, and will provide good approximations to and when both samples are large.) Small-Sample Confidence Interval for ( ) : Independent Samples ( ) 2 1 2 11 p xx t s  −± +   Where ( ) ( ) 112 2 2 2 p n sn s s +− = and t is based on ( ) 2 degrees of freedom
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Chapter Nine - Lecture Notes Chapter Nine: Inferences Based...

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