Statistical Sciences 2035 Lecture Notes - Interval Estimation, Standard Deviation, Statistical Process Control

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The Empirical Rule
If the population distribution is bell-shaped (normal distribution), symmetrical with a
single peak, with mean μ and standard deviation σ, then
1. 68.26% of all population units are within (plus or minus) one standard deviation
of the mean and thus lie in the interval [μ σ, μ + σ]
2. 95.44% of all population units are within (plus or minus) two standard deviations
of the mean and thus lie in the interval [μ 2σ, μ + 2σ]
3. 99.73% of all population units are within (plus or minus) three standard
deviations of the mean and thus lie in the interval [μ 3σ, μ + 3σ]
Graphically
Note
1. An interval that contains a specified percentage of the individual measurements in
a population is called a tolerance interval
2. Often we interpret the three-sigma interval [μ ± 3σ] to be a tolerance interval that
contains almost all of the measurements in a normally distributed population
3. Of course, we usually do not know the true values of μ and σ.
We must estimate the tolerance intervals by replacing μ and σ in these
intervals with the sample mean
x
and standard deviation s
We use
x
to estimate μ
Back to Example 2 .2
We saw that the distribution of waiting times for Dr. Kim was approximately normal.
We also calculated
x
= 14.28 minutes
s = 7.50 minutes
Thus,
14.28+- 7.5=6.78~21.78
To contain about 66.26% of the possible waiting time;
14.28 +-2*7.5=0~29.28
To contain about 95.44% of the possible waiting times;
14.28+-7.5*3=0~35.78
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