Six Sigma Training

1
Six Sigma Training

• Dr. Robert O. Neidigh
• Dr. Robert Setaputra

2
Variable Types Page 235

• Attribute Data a variable is either classified
  into categories or used to count occurrences of a
  phenomenon, also referred to as classification or
  categorical data. Examples gender, reasons for
  defects, and votes for candidates
• Measurement Data results from a measurement
  taken on an item or person of interest, also
  called continuous or variables data. Examples
  height, weight, temperature, and cycle time

3
Measures of Central Tendency

• Measures that try to describe or quantify the
  middle of a data set.

4
Measures of Central Tendency

• Mean average of all data points
• Median value such that at least half the data
  points are less than or equal to the value and at
  least half the data points are greater than or
  equal to the value
• Mode value in the data set that occurs most
  frequently
• First Quartile value such that at least 25 of
  the data points are less than or equal to the
  value and at least 75 of the data points are
  greater than or equal to the value
• Third Quartile value such that at least 75 of
  the data points are less than or equal to the
  value and at least 25 of the data points are
  greater than or equal to the value
• Minitab example on Page 249

5
Measures of Variation

• Measures that try to describe or quantify the
  amount of spread or variation in a data set

6
Measures of Variation

• Range distance from the smallest data point to
  the largest data point
• Variance and Standard Deviation measure of how
  much the data points fluctuate around the mean
• Minitab example on Page 249

7
What is standard deviation?

• Standard deviation is a measure of variation
  within a data set.
• The larger the standard deviation, the more
  variation in the data set and vice versa.
• Technically, standard deviation is a measure of
  variation about the mean.
• Roughly speaking, standard deviation is the
  average distance between each data point and the
  mean.
• Motivate measure of variation through examples of
  small data sets.
• Population divide by n, sample divide by n -
  1
• Show students Normal.xls file.

8
Continuous Probability Distributions

• Can assume an infinite number of values within a
  given range
• Probability of any one point is zero
• Probabilities are measured over intervals
• Area under curve defines probability
• Use calculus to calculate probabilities
• Ugh!!!
• Normal probability distribution is one type
• Fortunately, probabilities already calculated and
  contained in a table for normal distribution

9
Characteristics of Normal ProbabilityDistribution

1. Bell-shaped
2. Symmetrical
3. Mean, median, and mode are the same
4. Asymptotic tails never touch X-axis
5. Completely described by its two parameters
   mean(µ) and standard deviation(s)
6. There are an infinite number of possible normal
   probability distributions

10
How do we calculate probabilities?

• Since there are an infinite number of normal
  distributions, how can we possibly calculate
  probabilities for all of them? Fortunately, there
  is a unique characteristic of all normal
  distributions that allows us to do so. The
  probability of having a value above/below a point
  that is X standard deviations above/below the
  mean is the same for every possible normal
  distribution. The probabilities for the standard
  normal distribution (µ 0 and s 1) can be used
  for every other normal distribution. These
  probabilities can be found in the standard normal
  probability table. Our task is to convert every
  normal distribution to the standard normal, this
  is called standardizing.

11
How do we standardize?

• The distance between any point on our normal
  distribution of interest and the mean is found.
  We now want to put this distance in units of
  standard deviation, to do so we divide the
  distance between the point and the mean by our
  standard deviation. This value is called a
  Z-value and tells us how many standard deviations
  above or below the mean a point is. If the
  z-value is positive, the point is above the mean
  and if the z-value is negative the point is below
  the mean.
• Z-value (point minus the mean)/standard
  deviation
• The standard normal table always gives the
  probability of having a value less than the
  Z-value.

12
Finding the probability of having a value less
than a given point

• Find the Z-value for the given point
• The Z-value lets us know how many standard
  deviations above/below the mean the point is
• Look up the probability in the standard normal
  table
• This is the probability of having a value less
  than the given point
• µ 70 and s 10, find probability of having a
  value less than 66

13
(No Transcript)
14
(No Transcript)
15
Finding the probability of having a value greater
than a given point

• Find the Z-value for the given point
• The Z-value lets us know how many standard
  deviations above/below the mean the point is
• Look up the probability in the standard normal
  table
• This is the probability of having a value less
  than the given point
• Subtract this probability from one to find the
  probability of having a point greater than the
  given point
• µ 70 and s 10, find probability of having a
  value greater than 56

16
(No Transcript)
17
(No Transcript)
18
Finding the probability of having a value between
two points

• Find the Z-values for the given points
• The Z-values let us know how many standard
  deviations above/below the mean the points are
• Look up the probabilities in the standard normal
  table for the two Z-values
• These are the probabilities of having a value
  less than the given point associated with each
  Z-value
• Subtract the probability associated with the
  smallest Z-value from the probability associated
  with the largest Z-value
• This is the probability of having a value between
  the two points
• µ 70 and s 10, find probability of having a
  value between 57 and 76

19
(No Transcript)
20
(No Transcript)
21
Finding the point on a normal distribution
associated with a given probability

• Find the probability in the standard normal table
• Find the Z-value associated with the probability
• Convert the Z-value to a point on the normal
  distribution
• Mean plus (Z-value times standard deviation)
• µ 70 and s 10, find the value such that 70
  of the charge amounts will be greater than that
  amount

22
(No Transcript)
23
(No Transcript)
24
Sampling Methods

• Reasons for sampling
• Too time consuming to check entire population
• Too expensive to check entire population
• Sample results are adequate
• Destructive testing
• Impossible to check entire population

25
Sampling Definitions

• Simple random sample each item in the
  population has the same probability of being
  selected
• Sampling error difference between a sample mean
  and the population mean
• Sampling distribution of the sample mean
  probability distribution of all possible sample
  means of a given sample size
• Standard error of the mean standard deviation
  of the sampling distribution of sample means
  (average sampling error)

26
When is sampling distribution normal?

• If population distribution is normal, then
  sampling distribution is normal for any sample
  size
• If sample size is greater than or equal to
  thirty, then sampling distribution is always
  normal

27
Properties of normal sampling distribution?

• Sampling distribution mean (µx-bar) equals
  population mean (µ)
• Standard error (sx-bar) equals population
  standard deviation (s) divided by the square root
  of the sample size (n)
• Once we know the mean and standard error of the
  sampling distribution and we know it is normally
  distributed we are set to compute probabilities

28
Notation
29
Example

• Captain Ds tuna is sold in cans that have a net
  weight of 8 ounces.
• The weights are normally distributed with a mean
  of 8.025 ounces and a standard deviation of 0.125
  ounces.
• You take a sample of 36 cans.

30
Example Cont.
31
Example Cont.

• What is the probability of having a sample mean
  greater than 8.03 ounces?

32
(No Transcript)
33
(No Transcript)
34
Example Cont.

• What is the probability of having a sample mean
  less than 7.995 ounces?

35
(No Transcript)
36
(No Transcript)
37
Example Cont.

• What is the probability of having a sample mean
  between 7.995 ounces and 8.03 ounces?

38
(No Transcript)
39
(No Transcript)
40
Hypothesis Testing

• Hypothesis a statement about a population
  developed for the purpose of testing
• Hypothesis test a procedure based on sample
  evidence and probability theory to determine
  whether the hypothesis is a reasonable statement
• Key Point Anytime a decision is made about a
  population based upon sample data an incorrect
  decision may be made

41
Type I and Type II Errors

• Type I Error rejecting a true null hypothesis
• Type II Error accepting a false null hypothesis
• Unfortunately, in hypothesis testing the
  probability of a Type I Error (a) is inversely
  related to the probability of a Type II Error
  (ß). If we decrease the probability of a Type I
  Error, then the probability of a Type II Error
  increases and vice versa.
• What are Type I and Type II errors in the U.S.
  Legal System?