Class 3

Discuss write-up for assignment 1.

What you need to know from last time

Summary measures; covariance and correlation

Finance arithmetic, basic probability (assigned reading)

Capable process - meets engineers specs.

In control process - mean and variance stable over time

Todays class

The standard error of the mean;

The Central Limit Theorem

Tracking sample means and standard deviations: x-bar and s-charts. Setting control limits

Confidence intervals

Using a confidence interval to make a decision

Standard error of the mean

Sample means are less variable than raw data

SE( ) = where is the true s.d. of a single observation and n is the number of observations in the sample mean

The Central Limit Theorem

Sample means are approximately normally distributed. (see p.68 of CaseBook)

E( ) = .

Var( ) = .

s.d.( ) = SE( ) = .

Because sample means are approx. normal can use Empirical Rule on them.

Control charts

Two types

X-bar chart; track sample means

s-chart; track sample standard deviations

Setting the control limits - two ways (JMP gives choice);

From the engineer; use their specs to create limits

From the data; use overall sample mean and overall sample variance plus the Empirical Rule to create limits (typically 3-sigma)

Two examples

shaftxtr.jmp A well behaved process -- in control. p.70.

carseam.jmp A process that fails to meet engineers specs. p.82.

Notes

S-charts are usually one-sided in manufacturing

Dealing with miracles; someone has to win the lottery but the same person should not win it three times in a row. Take action on observing a rare event.

Daily means, weekly means, monthly means or WHAT? (p.80 of CaseBook) Better normality, less trawling and greater sensitivity (means pick up small changes faster -- up to a point).

Trawling through the data; the more things that are looked at the more likely we observe "false significance" (false positive - Type II error) pp.60,61 of CaseBook.

Confidence intervals

What is it?

1. A range of feasible values for an unknown population parameter, e.g. or

2. A statement conveying the confidence that the range of feasible values really does include the unknown population value

Where does it come from?

Inverting the Empirical rule

If 95% of the time the sample mean is within +/- 2 standard errors from , then 95% of the time the true is within +/- 2 standard errors from the sample mean

Why is it important?

Move away from a single "estimate" to a range of values, which is more realistic

Get to make the meta-level statement - our confidence about the first statement

How do I use it to make a decision?

Example, is 812 a feasible value for the true mean?

Answer: look to see if 812 lies in the confidence interval

If it's in the interval then it's a feasible value

If it's outside the interval then it is not feasible

shaftxtr.jmp A confidence interval for the population mean. p.99.

comppur.jmp A confidence interval for the intent to purchase. p.106.