Stat 601, Fall 2001, Class 2




What you need to know from last time

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Summary measures; mean, median,variance,sd,IQR
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Graphical summaries/diagnostics; histogram,boxplot,normal quantile plot
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If approx normal then can use empirical rule
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What is the Empirical rule?
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Often data is approx normal - but not always

Today's class

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Tracking sample means and standard deviations: x-bar and s-charts. Setting control limits
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The standard error of the mean; $\sigma/\sqrt{n}$
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The Central Limit Theorem
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Confidence intervals
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Using a confidence interval to make a decision
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Assumptions and their role in analysis
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Ideas behind sampling

Monitoring the mean and variance of a process

Example

ShaftXtr.jmp

Objective

Monitor a production process assuming observations are independent.

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Achieve this by placing control limits
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How to choose limits - can use empirical rule on sample means
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In control: mean and variance stable over time
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Capable: process meets specs
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E.R. needs to know s.d. of the sample means
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SD of ${\overline X} = \sigma/\sqrt{n}$ where n is number of observations in sample mean
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Can use overall sample mean +/- 3 $\sigma/\sqrt{n}$ as "3 sigma limits"
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Chances a particular observation is outside these limits if process is in control is 1 -.997 (from ER), ie small
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Unlikely evens signal something is wrong -> take action

Standard error of the mean

The Central Limit Theorem

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Sample means are approximately normally distributed. (see p.66 of CaseBook)
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E( $\overline{X}$) = $\mu$.
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Var( $\overline{X}$) = $\sigma^2/n$.
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s.d.( $\overline{X}$) = SE( $\overline{X}$) = $\sigma/\sqrt{n}$.
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Because sample means are approx. normal can use Empirical Rule on them.

Control charts

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Two types
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X-bar chart; track sample means
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s-chart; track sample standard deviations
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Setting the control limits - two ways (JMP gives choice);
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From the engineer; use their specs to create limits
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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.

CarSeam.jmp A process that fails to meet engineers specs.

CompChip.jmp A process that breaks down.

Notes

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S-charts are usually one-sided in manufacturing
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Dealing with miracles; someone has to win the lottery but the same person should not win it three times in a row. (p.63 of CaseBook)
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Daily means, weekly means, monthly means or WHAT? (p.79 of CaseBook)

Confidence intervals

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What is it?
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1. A range of feasible values for an unknown population parameter, e.g. $\mu$ or $\sigma^2$
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2. A statement conveying the confidence that the range of feasible values really does include the unknown population value
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Where does it come from?
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Inverting the Empirical rule
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If 95% of the time the sample mean is within +/- 2 standard errors from $\mu$, then 95% of the time the true $\mu$ is within +/- 2 standard errors from the sample mean
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Why is it important?
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Move away from a single ``estimate'' to a range of values, which is more realistic
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Get to make the meta-level statement - our confidence about the first statement
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How do I use it to make a decision?
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Example, is 812 a feasible value for the true mean?
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Answer: look to see if 812 lies in the confidence interval
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If it's in the interval then it's a feasible value
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If it's outside the interval then it is not feasible

ShaftXtr.jmp A confidence interval for the population mean.

CompPur.jmp A confidence interval for the intent to purchase.

Sampling

Introduction

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Context; there is a target population - the group you wish to make inferences about. You draw a sample. Use the sample to make statements about the population.
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Sample must be representative of the population
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Sampling is the way to obtain reliable information in a cost effective way (why not census?)
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Objective; collect information. Issues:
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What to measure?
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How accurate do we need it?
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How often do we need it?
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Does it meet end user requirements?

Issues in sampling

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Representativeness
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Interviewer discretion
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Respondent discretion - non-response
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Key question: is the reason for non-response related to the attribute you are trying to measure? Illegal aliens/Census. Start-up companies/not in phone book. Library exit survey.
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Good samples;
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Good samples; probability samples; each unit in the population has a known probability of being in the sample
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Simplest case; equal probability sample, each unit has the same chance of being in the sample
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Bad samples - the rest, convenience samples

The utopian sample for analysis

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You have a complete and accurate list of ALL the units in the target population (sampling frame)
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From this you draw an equal probability sample (generate a list of random numbers)
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Reality check; incomplete frame, impossible frame, practical constraints on the simple random sample (cost and time of sampling)

Precision considerations

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How large a sample do I need? p.117
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Focus on confidence interval - choose coverage rate (90%, 95%, 99%) margin of error (half the width). Typically trade off width against coverage rate.
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Simple rule of thumb for a population proportion - if it's a 95% CI then use n = 1/(margin of error)**2.

Examples

Survey1.jmp A hotel customer satisfaction survey.


Subsections

2001-09-06