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

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1 Dec 2011

School

Department

Professor

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