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Normal_Curves_z-scores

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Normal_Curves_z-scores

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Normal_Curves_z-scores

1. 1. NORMAL CURVE 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 µ +σ−σ +1 9 6. σ−1 9 6. σ Area = .6823 2/3 of the population lies between +- 1 st. deviation Area = .95 2.5% of population2.5% of population 95% of the population lies between +- 1.96 st. deviations
2. 2. NORMAL CURVE 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0 50 100 150 200 250 300 µ σ= =1 7 0 4 5, Area = .4406 Z x i = − = − = − ( ) ( ) . µ σ 1 0 0 1 7 0 4 5 1 5 6 Area = .5 - .4406 = .0594 Area under the curve = 1
3. 3. Three Distinct DistributionsThree Distinct Distributions Mean St. Dev. Mean St. Dev. 1) Population 2) Sample 3) Sampling Proportions Means Distributions N
4. 4. Central Limit TheoremCentral Limit Theorem If Repeated Random Samples of sizeIf Repeated Random Samples of size nn are drawn from any population (ofare drawn from any population (of whatever form) having a Meanwhatever form) having a Mean µµ andand Standard DeviationStandard Deviation , then as, then as nn becomesbecomes very largevery large, the Sampling Distribution of, the Sampling Distribution of sample means approachessample means approaches NormalityNormality withwith MeanMean µµ and Standard Deviationand Standard Deviation σ / .n
5. 5. % of Population that pays \$250 = .17% 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0 50 100 150 200 250 300 Z a r e a = − = = = 2 4 9 5 1 7 0 4 5 7 9 5 4 5 1 7 7 4 6 1 6 . . . . Z a r e a = − = = = 2 5 0 5 1 7 0 4 5 8 0 5 4 5 1 7 9 4 6 3 3 . . . . . . . .4 6 3 3 4 6 1 6 0 0 1 7 1 7 %− = o r µ σ= =1 7 0 4 5,
6. 6. 20% of the population pays less than \$132.20. 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0 50 100 150 200 250 300 area = .2 area = .3 Z-score of area of .2995 (~.3) = -0.84 − = − − = − ⋅ = − = − + = 0 8 4 1 7 0 4 5 1 7 0 8 4 4 5 3 7 8 3 7 8 1 7 0 1 3 2 2 . . . . . x x x µ σ= =1 7 0 4 5, Z x = − µ σ \$132.20
7. 7. Sampling Distribution for MeansSampling Distribution for Means ““If repeatedIf repeated Random Samples of SizeRandom Samples of Size nn areare drawn from a normal population withdrawn from a normal population with MeanMean µµ andand Standard DeviationStandard Deviation , the Sampling, the Sampling Distribution of sample means will be normalDistribution of sample means will be normal with Meanwith Mean µµ and Standard Deviationand Standard Deviation --known as the Standard Error.”--known as the Standard Error.”σ / n
8. 8. Sampling Distribution for Sample Means for a Population with 0 0.05 0.1 0.15 0.2 0.25 0.3 7000 7050 7100 7150 7200 7250 7300 7350 7400 7450 7500 7550 7600 7650 7700 7750 7800 7850 7900 7950 8000