Exam Thursday.

1

• Exam Thursday.
• See website tomorrow for more information on exam
  notes homework solutions practice problems
  solutions old exams.
• Today More about six sigma quality control
  more about the normal distribution
• First, an aside about a statistical look at a
  recent newspaper story

2

• Study Ties 6-7 Hours of Sleep to Longer Life (
  AP ) LEAD PARAGRAPH - Research suggests that
  adults live longer if they get six or seven hours
  of sleep a night rather than the accepted
  standard of eight hours.
• The research is based on a nationwide survey of
  1.1 million adults. It found that those who slept
  eight hours a night were 12 percent more likely
  to die within six years than those who got 6.5 to
  7.5 hours of sleep. The increased risk was more
  than 15 percent for those who reported getting
  more than 8.5 hours or less than about 4 hours
  nightly.

Can you live longer bysleeping less?
3
Perhaps theres a simpler explanation
Sickness
Sleep More
causes
causes
Observed relationwithout causality
Earlier Death
This is called confounding. The relationship
between sleep and earlier death is confounded by
sickness.
4
Standard normal distribution
Standard Normal Distribution Mean 0 Std dev
sigma 0
Normal PDF
0
-1
1
-2
2
5
In general, the x-axis on the normal pdf is in
terms of the mean and multiples of the standard
deviation (sigma)
Normal Distribution with x-axis in terms of
mean and standard deviation (sigma)
This re-labeling of the x-axis is what centering
and scaling is really doing
Normal PDF
mean
Mean minus 1 sigma
Mean plus 1 sigma
Mean minus 2 sigma
Mean plus 2 sigma
6

• The point is that normal probabilities and
  rareness can be described using standard
  deviations away from the mean
• Example (heights problem on homework)Is a 6
  2 (74) person rare?
• Problem gave that heights are normally
  distributed with mean 69 and standard deviation
  (sigma) 3.5.
• 74 69 1.433.5i.e. 74 average height
  1.43 sigma
• An event is rare if
• 1. Probability of seeing that event or something
  more extreme (something further from the mean) is
  small.
• Or if pdf is close to zero at the point
• Pr( hieght gt 74) Pr( normal random variable gt
  mean1.43 std dev) 1 Pr(Z lt 1.43) 1
  0.9236 0.0764
• So about 1 in 14 people are 62 or taller.

7
Probability meaning of 6 sigma

• Even if you shift the process mean for the center
  of the specifications to 1.5 standard deviations
  toward one of the specifications, then you will
  expect no more than 3.4 out of a million defects
  outside of the specification toward which you
  shifted.
• (I know its convoluted, but thats the
  definition)

8
What does 6 sigma mean?(example)

• Suppose a product has a quantitative
  specificationex Make the gap between the car
  door and the car body between 3.4 and 4.6mm.
• When cars are actually made, the std dev of car
  door gap is 0.1mm. i.e. X1,,Xn are gap widths.
  The sqrt(sample variance of X1,,Xn) 0.1mm

9
Statistically, six sigma means that Upper Spec
Lower Spec gt 12 sigma (i.e. Specs are fixed.
Lower the manufactuing process variability.)
Distribution of gap widths
Lower specification
Upper specification
Center of spec 4mm gap
Shifted mean 3.85mm gap
3.4mm
4.6mm
4.6 3.4 1.2 120.1 12sigma
Probability of beingout here is Pr( gap is less
than 3.4 ) Pr( (gap 3.85)/0.1 lt
(3.4-3.85)/.1) Pr( Z lt -4.5) 3.4/1,000,000
Arbitrary magic number for 6s
10
Probability meaning of 6 sigma

• In general
• Assume process mean is 1.5 standard deviations
  toward the lower spec i.e. E(X)4-1.5s and
  assume X has a normal distribution.
• When the process is in control enough so that
  the distance between the center of the specs and
  the lower spec is least 6s, then
• Pr(X below lower spec) Pr( Xlt4- 6s)Pr(X-
  (4-1.5s))/s lt (4-6s-(4-1.5s))/s Pr(Zlt-4.5)
  3.4/1,000,000

11
Control Charts

• Let X an average of n measurements.
• Each measurement has mean m andvariance s2.
• Fact
• By the central limit theorem, almost all
  observations of X fall in the interval m /-
  3s/sqrt(n) (i.e. mean /- 3 standard deviations)
• s/sqrt(n) is also called sx or standard error

12
Use the fact to detect changes in production
quality

• Idea let xi average door gap from the n cars
  made by shift i at the car plant

m3 s/sqrt(n) (Upper Control Limit)
x7
x6
x1
x8
x3
m
x2
x5
m-3 s/sqrt(n) (Lower Control Limit)
x4
shift
Points outside the /- 3 std error bounds, are
called out of control. They are evidence that
m and or s are not the true mean and std dev any
more, and the process needs to be readjusted.
Calculate the false alarm rate ( 26/10,000)
13
Assume 100 new people arepolled.Assume true
pr( a new person approves) 0.57.Let P P
hat number who approve/100 Whats an
approximation tothe distribution of P-hat?Use
the approximation todetermine a number so
thatthe Pr(p-hatgt that number) 0.95.
14
(No Transcript)
15

• Suppose true p is 0.57.
• If survey is conducted again on 100 people, then
  P-hat N(.57,(.57)(.43)/100) N(.57,
  0.002451)Want p0 so that Pr(P-hatltp0) 0.95
  Pr(P-hatltp0) 0.95 means Pr(Z lt
  (p0-.57)/0.0495) 0.95.Since Pr(Zlt1.645)
  0.95, (p0-.57)/0.0495 1.645 (p0-.57)
  0.0814 p0 0.6514