Statistical methods and
application
Evans’ chapter 11
Dr. Pham Huynh Tram
phtram@hcmiu.edu.vn
When to use..
In Quality control and improvement, to analyze a process
-
describe characteristics of a product or process
conclude parameters of a product or process
Examples
→ sample of the weight of
38.1
38.5
38.3
37.3
38.4
39.2
38.9
38.7
39
38.6
38.1
38.5
38.4
37.6
38.4
39.2
38.9
38.7
39
38.6
castings (in kilograms)
38.2
38.5
38.4
37.7
38.4
39.3
38.9
38.7
39.1
38.7
from an production line
38.2
38.5
38.4
37.8
38.4
39.3
38.9
38.7
39.1
38.7
in the Harrison
38.3
38.6
38.4
37.9
38.4
39.3
38.9
38.7
39.1
38.7
Metalwork foundry.
38.3
38.6
38.4
37.9
38.4
39.4
39
38.7
39.1
38.7
38.3
38.6
38.4
37.9
38.4
39.4
39
38.7
39.1
38.7
38.3
38.6
38.4
38.1
38.4
39.5
39
38.8
39.1
38.7
38.3
38.6
38.4
38.1
38.5
39.6
39
38.8
39.2
38.7
38.3
38.6
38.4
38.1
38.5
39.9
39
38.8
39.2
38.7
What would you do to analyse
the data?
What do you conclude from your
analysis?
Descriptive Statistics
Mean
38.6320
Standard Error
0.0444
Median
38.6000
Mode
38.4000
Standard Deviation
0.4436
Sample Variance
0.1967
Range
2.6000
Minimum
37.3000
Maximum
39.9000
Sum
3863.2000
Count
100.0000
→ data are fairly normally distributed, with some slight skewing to the right.
Examples
A soup processed by our company for the month of July. Output in the month of July of brand A
soup is 50,000 cans.
We want to determine the average weight of cans of brand and so randomly select 500 cans of
brand A soup from the July output. We get the average weight of 500 cans is 295g
Next, we want to test the validity of a claim that the average weight of the cans is no less than
300g. Is this measured average of 295g is significantly smaller than the claimed mean of 300g?
Recall:
What are “population”, “sample”, “parameter”, “statistic”, “estimator”
descriptive statistics, inferential statistics
Statistical Methods
-
Clarify
characteristics of
a process
-
predict future
results
6
-
-
Quantify the
uncertainty of
sample data
Test factor
significance
-
Identify quality
problems
-
Means of
measuring
improvement
Statistical Foundations
✹ Random variables
Discrete vs continuous
✹ Probability distributions
A theoretical model of the relative frequency of a random
variable
✹ Sampling
Form a basis for application of statistics
Random variables and Probability distribution
● Binomial: # of successes in n trials
● Negative binomial: # of trials for r successes
● Geometry: #of trials for 1st success
● Poisson: # of independent events that occur in a fixed amount of
time or space
● Normal: distribution of a process that is the sum of a number of
component processes. Eg. assembly
Recall: f(x) probability density function (pdf)/ p(x) probability mass function (pmf)
F(x) cumulative distribution function (cdf)
Binomial
Poisson
Normal
Examples
1. A process is known to have a nonconformance rate of 0.02. If a random
sample of 100 items is selected, what is the probability of finding 3 nonconforming items?
2.A process is known to produce about 6% nonconforming items. If a random
sample of 200 items is chosen, what is the probability of finding between 6 and 8
nonconforming items?
3. New Orleans Punch was made by Frutayuda, Inc. and sold in 16-ounce cans to benefit
victims of Hurricane Katrina. The mean number of ounces placed in a can by an automatic fill
pump is 15.8 with a standard deviation of 0.12 ounce. Assuming a normal distribution, what
is the probability that the filling pump will cause an overflow in a can, that is, the probability
that more than 16 ounces will be released by the pump and overflow the can?
4. Georgia Tea is sold in 2 liter (2000 milliliter) bottles. The standard deviation for the filling
process is 15 milliliters. If the process requires a 1 percent, or smaller, probability of overfilling, defined as over 1990 milliliters, what must the target mean for the process be?
5. Outback Beer bottles have been found to have a standard deviation of 5 ml. If 95 percent of
the bottles contain more than 230 ml, what is the average filling volume of the bottles?
Sampling
● Sampling forms the basis for applications of statistics
● Suppose that we want to determine the attitudes of students about
the quality of care they received in IU. Several factors should be
considered before making this study:
-
What is the objective of the study
-
What type of sample should be used?
-
What possible error might result from sampling?
-
What will the study cost?
13
Sampling Plan
A good sampling plan should select a sample at the
lowest cost that will provide the best possible
representation of the population, consistent with the
objectives of precision and reliability that have been
determined for the study.
14
Factors to consider
✹ Sample
size
✹ Appropriate sample design
15
Sampling Error
● Sampling error (statistical error)
Occurs naturally, for a sample may not always be representative of the population no
matter how carefully it is selected
→ reduce this error by ?
● Nonsampling error (systematic error)
Bias: tendency to systematically over or underestimate true values
Non-comparable data: data that com from 2 populations
Uncritical projection of trends: assumption that what happened in the past will continue
into the future
Causation: assumption that because 2 variables are related, one must be the cause of
changes in the other
Improper sampling: use of erroneous method for gather data, thus biasing results
→ reduce this error by ?
16
Sampling Error
✹ Sampling error (statistical error)
✹ Nonsampling error (systematic error)
17
Sampling Methods
● Simple random sampling
○
Every item in the population has an equal probability of being selected
● Stratified sampling
○
The population is partitioned into groups, or strata, and a sample is selected from each
stratum
● Systematic sampling
○
Every nth (4th, 5th, etc.) item is selected
● Cluster sampling
○
divide a population into smaller groups known as clusters. Randomly select among
these clusters to form a sample
Example
● A box of 1000 plastic components for electrical connectors is thoroughly mixed
and 25 parts are selected randomly without replacement
● A particular nursing unit has 30 patients. Five patient records are to be sampled to
verify the correctness of a medical procedure
● A population of 28,000 items is produced on 3 different machines
Machine 1: 20,000 items
Machine 2: 5,000 items
MAchine 3: 3,000 items
Assume that a specific confidence level requires a sample of 525 units. A simple
random sample of 250 units from a machine 1, 150 units from machine 2, 125
units from machine 3 are taken
Example
● A population has 4000 units and a sample of size 50 is required. Select the
first unit randomly from among the first 80 units. Every 80th (4000/50) item
after that should be selected
● Products are boxed in groups of 50. We draw a sample of boxes and inspect
all units in the boxes selected
Central Limit Theorem
✹ If simple random samples of size n are taken from any population, the
probability distribution of sample means will be approximately normal
as n becomes large.
The mean of the sample
means for this probability
distribution will approach
µ, and the standard
deviation
of
the
distribution will be σ /
sqroot(n)
The CLT is extremely
important in any SQC
techniques that require
sampling.
(sampling
distribution)
Sample size
Simple random sampling is generally used to estimate population parameters such as means, proportions, and
variances. Consider the sample size when using sample mean to provide a point estimate of the population
mean for variables data. A 100(1 - 𝛼)% confidence interval on sample mean is given by
→ sampling error E:
This sample size (n) will provide a point
estimate having a sampling error of E or
less at a confidence level of 100(1 - 𝛼)%.
A preliminary sample or a good guess
based on prior data or similar studies can
be used to estimate 𝞼
Determine the sample size for estimating a population proportion for
attributes data.
Example - sampling
● A Firm conduct a process capability study on a critical quality dimension
wishes to determine the sample size required to estimate the process mean
with a sampling error of at most 0.1 at a 95% C.I. level. From control chart
data, an estimate of a standard deviation of the process was found to be 0.47.
Find the appropriate sample size
● A sample from a large finished group inventory is needed to determine the
proportion of noncomforming product. Historically, about 0.5% level of noncomformance has been observed. A 90% confidence level with an allowable
error of 0.25% is desired. Find the appropriate sample size
Example - sampling
● Localtel, a small telephone company, interviewed 150 customers to determine
their satisfaction with service. 27 expressed dissatisfaction. If an allowable
error for the proportion dissatisfied is 0.05, is the sampling size sufficient
with 90% confidence level?
● A management engineer at XYZ hospital determined that she needs to take a
work sampling study to see whether the proportion of idle time in the
diagnostic imaging department had changed since being measured in a
previous study several years ago. At that time, the percentage of idle time
was 10%. If the engineer can only take a sample of 800 observations due to
cost factors and can tolerate an allowable error of 0.02, what percent
confidence level can be obtained from the study?