# 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 What is the probability that signal will be a false alarm? By the empirical rule, the probability that a point lies within 3σ of the mean is 0.997. The probability that signal 1 will be a false alarm is 1 - 0.997 = 0.003. Remember, a false alarm means that the x distriution is really on the target distribution, and we simply have a very rare (probability of 0.003) event. | ## Out-of-Control Signal II: A run of nine consecutive points on one side of the center line (the line at target value µ) To find the probability that signal 2 is a false alarm, we observe that if the x distribution and the target distribution are the same, then there is a 50% chance that the x values will lie above or below the center line at µ. Because the samples are independent, the probability of a run of nine points on one side of the center line is (0.5)^9 = 0.002. If we consider both sides, this probability becomes 0.004. Therefore, the probability that signal 2 is a false alarm is approximately 0.004. | ## Out-of-Control Signal III: At least two of three consecutive points lie beyond the 2σ level on the same side of the center lineTo 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. |

## 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.