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.

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.