ch6 notes

# ch6 notes - Assumptions Robustness and Conditions for Valid...

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Assumptions, Robustness, and Conditions for Valid CIs & T-Tests T-tests and CIs are based on the assumption that the population values being studied have a Normal distribution. In reality, populations may be anywhere from slightly non-normal to very non-normal. Robustness of the T-procedures The T-test and CI are called robust to the assumption of normality because p-values and confidence levels are not greatly affected by violations of this assumption of normally distributed populations, especially if sample sizes are large enough. Conditions for a Valid 1-Sample CI and T-Test The data are a SRS from the population. The population must be large enough (at least 10 times larger than the sample size). Conditions for the sample: o If n is small (n < 15), the data should not be grossly non-normal or contain outliers. o If n is “medium” (15 n < 40), the data should not have strong skewness or outliers. o If n is large (n 40), the T -procedures are robust to non-normality. Checking if the conditions are met in your sample Always make a plot of the data to check for skewness and outliers before relying on T-procedures in small samples. Population 1 μ 1 Population 2 2 SRS of n histogram of all possible values for 12 x

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Comparing Two Independent Samples The sampling distribution for … 1 X is approximately Normal with Mean = 1 μ , SD= 1 1 n σ 2 X is approximately Normal with Mean = 2 , SD= 2 2 n 1 XX 2 is approx Normal with Mean = 12 , SD= 22 nn + ( ) () s SD z −− = Æ ( ) ( ) s SE t = Example : 30 mice given one of two diets for 21 days. Reduction in cholesterol in mg/dl is measured on day 22 Diet N Mean StDev Bean 15 26.46 5.90 Oat 15 32.23 9.56 12 0 : o HD µ −= 0 s xx xx D t SE = = Æ 0 2 11 s p x xD t s = ⎛⎞ + ⎜⎟ ⎝⎠ with 2 df n n = +− (Equal Variance T -test) Rejection Region at the α level of significance: Reject Ho in favor of Æ ' 0 s x t ss = + (Unpooled or Unequal Variance T -test) Satterthwaite’s approximation for the degrees of freedom: 2 (1 ) df + = ⎛⎞ ⎛⎞ ⎜⎟ ⎜⎟ ⎝⎠ ⎝⎠ + ) 2 | t t t / : : :| as i f t i f t i f t α µµ >≥ <≤ −≠ ≥
22 12 : o H σσ = : a H Fmax Test : (a.k.a. Hartley’s test) 2 max max, 2 min s s F s

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ch6 notes - Assumptions Robustness and Conditions for Valid...

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