Normal_Curves_z-scores

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. 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. Three Distinct DistributionsThree Distinct Distributions
Mean St. Dev. Mean St. Dev.
1) Population
2) Sample
3) Sampling
Proportions Means
Distributions
N

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. % 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. 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. 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drawn from a normal population withdrawn from a normal population with MeanMean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. 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