Lessons 6.1 and 6.2
Normal curves and Sampling Distributions
Properties of a normal curve
1. The curve is bell-shaped, with the highest point over the mean µ.
2. The curve is symmetrical about a vertical line through µ.
3. The curve approaches the horizontal axis, but never touches or crosses it.
4. The inflection (transition) points between cupping upward and downward occur above µ+σ and µ-σ.
5. The area under the entire curve is 1
µ and σ
a. Do these distributions have the same mean? if so what is it?
1. The means are the same, since both graphs have the high point over 6. µ = 6
b. One of the curves corresponds to a normal distribution with σ = 3 and the other to one
with σ = 1. Which curve has which σ?
1. Curve A has σ = 1 and Curve B has σ = 3. (Since Curve B is more spread out, it has the
larger σ value)
Empirical Rule
For a distribution that is symmetrical and bell-shaped (in particular, for a normal distribution):
Approximately 68% of the data values will lie within 1 standard deviation on each side of
the mean.
Approximately 95% of the data values will lie within 2 standard deviations on each side
of the mean
Approximately 99.7% (or almost all) of the data values will lie within 3 standard
deviations on each side of the mean.
Control Charts
Procedure
How to make a control chart for the random variable x is a plot of observed x values in
time sequence order.
1. Find the mean µ and standard deviation σ of the distribution by:
a. Using past data from a period during which the process was "in control"
or
b. Using specified "target" values for µ and σ.
2. Create a graph in which the vertical axis represents x values and the horizontal axis
represents time.
3. Draw a horizontal line at height µ and horizontal, dashed control-limit lines at µ +-
2σ and µ +- 3σ.
4. Plot the variable x on the graph in time sequence order. Use line segments to
connect the points in time sequence order.
Out-of-Control Signal I :One point falls beyond the 3σ line
Out-of-Control Signal II: A run of nine consecutive points on one side of the center line (the line at target value µ)
Out-of-Control Signal III: At least two of three consecutive points lie beyond the 2σ level on the same side of the center line
To determine the probability that signal 3 will produce a false alarm, we use the empirical rule. By this rule, the probability that an x value will be above the 2σ level is about 0.023. Using the binomail probability distribution, the probability of two or more successes out of three trials is:
See pg. 279
Taking into account both above and below the center line, it follows that the probability that signal 3 is a false alarm is about 0.004.
Z-scores
Z-value or Z-score(aka standard score) gives the number of standard deviations between the original measurement x and the mean µ of the x distribution.
z = x-µ/σ
A z-score close to zero tells us the measurement is near the mean of the distribution.
A positive z-score tells us the measurement is above the mean.
A negative z-score tells us the measurement is below the mean.
How to use a Left-tail style standard normal distribution table
1. For areas to the left of a specified z value, use the table entry directly.
2. For areas to the right of a specified z value, look up the table entry for z and subtract the area from 1.
3. For areas between two z-values, z1 and z2 ( where z2>z1), subtract the table area for z1 from the table area for z2.
Convention for using table 5 of Appendix II
1. Treat any area to the left of a z value smaller than -3.49 as 0.000.
2. Treat any area to the left of a z value greater than 3.49 as 1.000.