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STAT 513 Note - STAT 513 Note
Statistical Quality Control (Purdue University)
Studocu is not sponsored or endorsed by any college or university
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Chapter 1 Quality Control in the Modern Business Environment
1.1 The Meaning of Quality and Quality Improvement
1.1.1 Eight Dimensions of Quality
1. Performance (Will the product do the intended job?)
2. Reliability (How often does the product fail?)
3. Durability (How long does the product last?)
4. Serviceability (How easy is it to repair the product?)
5. Aesthetics (What does the product look like?)
6. Features (What does the product look like?)
7. Perceived Quality (What does the product look like?)
8. Conformance in Standards (What does the product look like?)
 Definition of Quality
o Traditional definition: fitness for use
 Two general aspects:
 Quality of design: variations in grades or levels of quality are intentional
 Quality of conformance: how well product conforms to specifications
required by the design
o Modern definition: Quality is inversely proportional to variability in the important
characteristics of a product (implies: ↓variability, ↑quality)
Fig1.1 warranty costs  quality; Fig1.2 distribution  variability
Japanese-produced transmissions having much lower costs (Japan: higher
quality)
 US: ~75% width of specifications; Japan: ~25% width of specifications (Japan:
lower variability)
 Japan: Less variability in the important characteristics  lower costs (better
quality)
 US: higher variability, lower cost (lower quality)
 ↓variability, ↓costs (↑quality)
(Modern) definition of quality improvement: reduction of variability in processes and products
o Equivalent definition: elimination of waste
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1.1.2
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Quality Engineering Terminology
Quality characteristics: parameters
Critical-to-quality (CTQ) characteristics:
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o Physical: length, weight, voltage, viscosity
o Sensory: taste, appearance, color
o Time orientation: reliability, durability, serviceability
Variability – statistical terms: classify data on quality characteristics as either attributes or
variable data
o Variable data: continuous measurements (ex. Length, voltage, or viscosity)
o Attribute data: discrete data (often taking form of counts)
Specifications
o For manufactured product, the specifications are designed measurements for quality
characteristics of components and subassemblies that make up he product, as well as
the designed value for quality characteristics in final products
o Lower specification limit (LSL): smallest allowable value for a quality characteristic
o Upper specification limit (USL): largest allowable value for a quality characteristic
o Target/ nominal value: a value of a measurement that corresponds to the desired value
for that quality characteristic
Defective or nonconforming product/ defect or nonconformity
o Nonconforming products are those that fail to meet one or more of their specifications
o A specific type of failure is called a nonconformity
o A nonconforming product is considered defective if it has one or more defects
o Defects are nonconformities that are serious enough to significantly affect the safe or
effective use of products
1.3 Statistical Methods for Quality Control and Improvement
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Fig 1.3: a process as a system with a set of inputs and an output
In a manufacturing process, controllable input factors x1, x2, …, xp are process variables such as
temperature, pressures, and feed rates; the input factors z1, z2, …, zq are uncontrollable (or
difficult to control inputs such as environmental factors or properties of raw materials provided
by an external supplier
The production process transforms the input raw materials, component parts, and
subassemblies into a finished product that has several quality characteristics
The output variable y is quality characteristic – that is, a measure of process and product quality
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Statistical and engineering technology useful in quality improvement. Specifically three major
areas:
o Statistical process control
o Design of experiments
o Acceptance sampling
Statistical process control (SPC):
o Control chart: one of the primary techniques of SPC
o Useful in monitoring processes, reducing variability through elimination of assignable
causes
o
Plots averages of measurements of a quality characteristic in sample taken from process
vs. time (or sample number)
Upper control limit (UCL)
Center line (CL)
Lower control limit (LCL)
Center line: represents where this process characteristic should fall if there are no
unusual sources of variability present
Designed experiment:
o Extremely helpful in discovering the key variables influencing the quality characteristics
of interest in the process
o An approach to systematically varying the controllable input factors in the process and
determining the effect these factors have on the output product parameters
o Invaluable in reducing the variability in the quality characteristics and in determining the
levels of the controllable variables that optimize process performance
o Factorial design: One major type of designed experiment
 factors are varied together in such a way that all possible combinations of factor
levels are tested
o
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T: target value
X1 low, x2,x3 high
X1 low, x2,x3 low
X1 high, x2,x3 high
X1 high, x2,x3 high
Fig 1.5a: two factors, each two levels, low and high  four possible test
combinations
 Fig 1.5b: three factors, each two levels, low and high  eight possible test
combinations
 Eg. ① measure x1 from low to high  increase average level of process output/
shift off target (T); ② when x2 and x3 high levels  process variability seem to
be substantially reduced
Acceptance Sampling
o Definition: inspection and classification of a sample of units selected at random from a
larger batch or lot and the ultimate decision about disposition of the lot, usually occurs
at two points: incoming raw materials or components, or final production
o Closely connected with inspection and testing of product
o One of the earliest aspects of quality control
o Outgoing inspection & incoming inspection
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Fig 1.6a: outgoing inspection
 Inspection operation is performed immediately following production, before the
product is shipped to the customer
Fig 1.6b: incoming inspection
 A situation in which lots of batches of products are sampled as they are received
from the supplier
Fig 1.6c: disposition of lots
 Sampled lots may either be accepted or rejected
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If accepted: ship to customers
If rejected:
o scrapped/recycled
o rework
o replaced with good units  rectifying inspection
1.4 Management Aspects of Quality Improvement
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Six sigma: effective quality management system
Logic of six sigma (textbook page 28-32)
Main idea: reduce variability (σ: standard deviation) in critical-to-quality (CTQ) variables to level
at which failure or defects are extremely unlikely
Recall: CTQ variables for a product are parameters/elements that jointly describe what
users/consumers think of as quality
o Smallest/largest value for CTQ variables: lower/upper specification limits (LSL/USL)
Formal description: reduce σ so that LSL/USL are at least 6σ from the mean
Parts per million (ppm) defective
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Fig 1.12a: shows a normal probability distribution as a model for a quality characteristic
with specification limits at three standard deviation (σ) on either side of mean.
The probability of producing a product within these specifications is 0.9973, 2700 parts
per million (ppm) defective
“three sigma quality performance”
 If a product consists of 100 independent components, then the probability that
any specific unit of product is non-defective is (0.9973)^100 = 0.7631
 23.7% products defective – not an acceptable situation
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The Motorola Six Sigma concept is to reduce the variability in the process so that the
specification limits are at least six standard deviations from the mean.
o The probability of producing a component within L/USL is 0.999999998 – one
component
o Probability that any specific unit of product is nondefective is (0.999999998)^100 or 0.2
ppm (100 independent components in a product)
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The process mean was still subject to disturbances that could cause it to shift by as much
as 1.5 standard deviations off target
o Under this scenario, a Six Sigma process would produce about 3.4 ppm defective
Why “Quality Improvement” is important: A simple example
o A visit of a fast-food restaurant: Hamburger (bun, meat, special sauce, cheese, pickle,
onion, lettuce, and tomato), fries, and a soft drink
 This product has ten components
o Is 99% good quality satisfactory?
o If we assume that all ten components are independent, the probability of a good meal
is: P{Single meal good} = (0.99)^10 = 0.9044
o Suppose family of 4, once a month:
 P{All meals good} = (0.9044)^4 = 0.6690
 P{All visits during the year good} = (0.6690)^12 = 0.0080
 Unacceptable!
o What if 99.9% good quality? Is it satisfactory?
 P{Single meal good} = (0.999)^10 = 0.9900
 P{All meals good} = (0.9900)^4 = 0.9607
 P{All visits during the year good} = (0.9607)^12 = 0.6186
o
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Chapter 2 The DMAIC Process
2.1 Overview of DMAIC
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DMAIC is a structured problem-solving procedure widely used in quality and process
improvement.
o Consisting of the following steps:
 Define
 Measure
 Analyze
 Improve
 Control
o It is usually associated with six sigma, but I can be used with any business or process
improvement effort
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Tollgates:
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Tollgates between each of major steps
Project is reviewed to ensure that this is on track, evaluate whether team can
successfully complete project on schedule
o Tollgates also present an opportunity to provide guidance regarding the use of specific
technical tools and other information about the problem
o Tollgates are critical to the overall problem-solving process; It is important that these
reviews be conducted very soon after the team completes each step.
Projects:
o Essential part of DMAIC (quality and process improvement)
o Breakthrough opportunity
o Financial systems integration
o Value opportunity of a project must be very clear
o Project selection
 Completed within a reasonable time frame
 Real impact on key business metrics
 Understand interrelationships and develop appropriate performance measures
What should be considered when evaluating proposed project?
o Suppose a company is operating at the 4σ level (6210 ppm defective, assuming the 1.5σ
shift in the mean that is customary with Six Sigma applications)
o Objective: achieve the 6σ performance level (3.4 ppm)
o Suppose that the criterion is a 25% annual improvement in quality level.
o Then to reach the Six Sigma performance level, it will take x years, where x is the
solution to this:  x is about 26 years
 Not going to work!
o Annual improvement
 50%  x=11 years
 75%  x=5 years
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2.2 The Define Step
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Objective:
o identify project opportunity
o verify or validate it represents legitimate
o must be important to customers (voice of the customer) and important to the business
project charter: a short document; a description of project and it scope & others
o a description of the project and its scope, the start and the anticipated completion
dates, an initial description of both primary and secondary metrics that will be used to
measure success and how those metrics align with business unit and corporate goals,
the potential benefits to the customer, the potential financial benefits to the
organization, milestones that should be accomplished during the project, the team
members and their roles, and any additional resources that are likely to be needed to
complete the project.
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Graphic aids:
o process map and flow charts
o value stream maps
o SIPOC diagram (a high level map of process)
 Suppliers
 Input
 Process
 Output
 Customer
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The Define Tollgate
o Symptoms (focus) not on possible causes or solutions
o Stakeholders
o Value opportunity
o Scope neither too small nor too large
o SIPOC diagram
o Obvious barriers or obstacles
o Action plan
2.3 The Measure Step
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Purpose: evaluate and determine present process state (evaluate and understand the current
state of process)
o Identify key process input variables (KPIV) and key process output variables (KPOV)
o Data: from historical records; from sampling; from observational studies
o Histograms, box plots, scatter diagrams, stem-leaf diagrams
o In transaction and service business, measurement system
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the capability of the measurement system should be evaluated.
(capacity/uniformity ↑, σ↓)
The Measure Tollgate
o Comprehensive process flow chart or value stream map
o KPIVs and KPOVs
o Measurement system capability documented
o Assumption noted
o Respond to requests
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2.4 The Analyze Step
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Purpose: use data from the Measure step to determine “cause-and-effect” relationships in the
process
Sources of variability:
o Common cause: embedded in system or process itself
o Assignable cause: arise from an external source
Tools:
o control charts
 separate common cause variability from assignable cause variability
o hypothesis testing and confidence intervals
 hypothesis testing: determine if different conditions of operations process
statistically significantly different results
 CIs: provide information about the accuracy with which parameters of interest
have been estimated
o regression models
 allows models relating outcome (effect) variables of interest to independent
input (cause) variables to be built
The Analyze Tollgate
o Opportunity targeted in the Improve step
o Data and analysis
o Other opportunities
o Track
2.5 The Improve Step
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Tools:
o Process redesign to improve work flow and to reduce bottlenecks
o Mistake-proofing
o Statistical tools: particularly designed experiments are probably most important
statistical tools in the Improve step
 Physical process or a computer model of process
Pilot test: a form of conformation experiment
o Evaluates and documents solution and confirms solution attains the project goals
o Iterative activity
The Improve tollgate
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How problem solution was obtained
Alternative solutions
Pilot test
Plans to implement pilot test results
Any risks of implementing solution
2.6 The Control Step
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Objectives:
o Complete all remaining work on project
o Provide the process owner with a process control plan
The process owner should be provided with before and after data on key process metrics,
operations and training documents, and updated current process maps.
The process control plan should be a system for monitoring the solution that has been
implemented, including methods and metrics for periodic auditing.
Control charts are an important statistical tool used in the Control step of DMAIC; many process
control plans involve control charts on critical process metrics.
o Control charts: an important statistical tool in control step
Transition plan: new process weight include a validation step (The transition plan for the process
owner should include a validation check several months after project completion.)
The Control tollage
o Data (before and after) available
o Process control plan & control charts
o Documentation
o Summary of lessons
o Opportunities not pursued in project
o Opportunities use results of project in other parts of business
Chapter 3 Modeling Process Quality
3.1 Describing Variation
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Stem-and-Leaf Plot
o Data: x1, x2, …, xn; each number xi consist of at least two digits, divide x1 into two parts
 A stem: one or more leading digits
 A leaf: two remaining digits
o For example, data from 10 to 100, number 76  divide value 76 into stem 7 and leaf 6
o Percentiles of data:
 the 100 kth percentile is a value such that at least 100 k% of the data values are
at or below this value and at least 100 (1 − k)% of the data values are at or above
this value
 The fiftieth percentile of the data distribution is called the sample median x-bar
 First, sort the observations in ascending order (or rank the data from
smallest observation to largest observation).
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Then the median will be the observation in rank position [(n − 1)/2 + 1]
on this list. If n is even, the median is the average of the (n/2)st and (n/2
+ 1)st ranked observations.
 Eg. n =11  x6; n=40  average of x20 and x21
The tenth percentile is the observation with rank (0.1)(40) + 0.5 = 4.5 (halfway
between the fourth and fifth observations)
The first quartile is the observation with rank (0.25)(40) + 0.5 = 10.5 (halfway
between the tenth and eleventh observation)
the third quartile is the observation with rank (0.75)(40) + 0.5 = 30.5 (halfway
between the thirtieth and thirty-first observation)
the interquartile range: IQR = Q3 − Q1
 occasionally used as a measure of variability
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plot of data in time order
take time order of observations into account
important factors to variability in quality improvement
plot data value versus time
time is an important source of variability in this process
processing cycle time for first 20 claims is substantially longer than processing
cycle time for second 20 claims
Histogram
o A more compact summary of data
o To construct a histogram for continuous data:
 Divide range of data into intervals called bins
 Sort data into bins, count number of obs. in each bin
 Use horizontal axis to represent measurement scale vertical scale to represent
counts or frequency or relative frequencies (frequencies in each bin divided by
total number of observations (n)
o Example 3.2 Metal Thickness in silicon wafers
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100 observations; # bins = sqrt(100)=10
Midpoint of first bin is 415Å
Histogram only has 8 bins containing nonzero frequency
Gives a visual impression of shape of the distribution of measurement and inherent
variability in data
Shows reasonably symmetric or bell-shaped distribution of metal thickness data
Histogram relatively sensitive to choice of number and width of bin for small data sets
Histogram suited for larger data sets, 75 to 100 or more observations
Height of each bin represents number of observations less than or equal to upper limits
of bin
 Eg. about 75 of 100 wafer less than 460 Å
To construct a histogram for discrete or count data (sample space is countable)
 Determine frequency (r relative frequency) for each value of x, each x value
corresponds a bin
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Plot frequency (or relative frequency) on vertical scale and values of x on
horizontal scale
Example 3.3: defects in automobile hoods
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o Proportion of hoods within at least 3 defects = 39/50 = 0.78
o Proportion of hoods with between 0 and 2 defects = 11/5 = 0.22
Numerical summary of data
o Sample average:
 Eg: table 3.2:
o Sample variance: variability in sample data is measured by sample variance
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 Larger sample variance, greater variability in data
o Sample deviation: square root of sample variance
 A measure of variability
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 primary advantage of sample standard deviation: expressed in original units of
measurements
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eg: metal thickness data  s^2 = 180.2928 Å  s = 13.03 Å
Box plot
o A graphic display; important features of the data, such as location or central tendency,
spread or variability, departure from symmetry, and identification of observations that
lie unusually far from the bulk of the data (these observations are often called
“outliers”).
o Display three quartiles, minimum and maximum of data on a rectangular box
o Example 3.4: Hole diameter
Q1= 0.25*12+0.5=3.5 (halfway between 3rd and 4th): (120.3+120.4)/2=120.35
Q2/ fiftieth percentile/ median = 120.6
Q3= .75*12+0.5=9.5 (halfway between 9th and 10th): (120.9+120.9)/2=120.9
Min=120.1
Max=121.3
A line at either end extends to the extreme values.
 Called whiskers
o Box plot – hole diameter distribution is not exactly symmetric – left/right boxes around
median not same
Comparative box plots – useful in graphical comparisons among data sets
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Too much variability at plant2
Plant 2 and 3 need to raise their quality index performance
Probability distributions
o Sample: a collection of measurements selected from some larger score or population
 Eg. analyze sample layer thickness data
o Probability distribution: mathematical model relates value of variable with probability of
occurrence of that value in the population
 Eg. layer thickness – a random variable
 Probability of occurrence of any value of layer thickness in population
Definition
o Continuous distributions
 When the variable being measured is expressed on a continuous scale, its
probability distribution is called a continuous distribution.
 The probability distribution of metal layer thickness is continuous
o Discrete distributions
 When the parameter being measured can only take on certain values, such as
the integers 0, 1, 2, . . . , the probability distribution is called a discrete
distribution.
 For example, the distribution of the number of nonconformities or defects in
printed circuit boards would be a discrete distribution.
Probability mass function (discrete variable)
o Discrete distribution; a series of vertical spikes, height of each spike proportional to the
probability
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Probability that the random variable x takes on the specific value xi as:
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Probability density function (continuous variable)
o A smooth curve
o Area under the curve equal to probability
o Probability that x lies in the interval from a to b is written as:
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Example 3.5: A discrete distribution
o Semi-conductor clips per day
o On average, 1% chips not conform to specifications
o Random sample 25 chips
o Let x random variable – number of nonconforming chips in sample
o Q1: probability distribution of x
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 Discrete distribution: binomial distribution
o Q2: probability of finding one or fewer nonconforming parts in sample
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Example 3.6: A continuous distribution
o Suppose x random variables – actual contents in ounces of a 1 pound bag of coffee
beans
o Probability distribution of x:
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 Range of x is interval [15.5, 17.0]
 Uniform distribution
Area under function f(x) – probability
Q: probability pf a bag containing less than 16.0 oz
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Expected value/ mean μ of a probability distribution is a measure of central tendency in the
distribution, or its location
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Suppose p(xi)=1/N, equation 3.5b
o Mean not necessarily 50th percentile of distribution (median)
o Mean not necessarily most likely value of random variable (mode)
Scatter, spread, or variability in a distribution
o Variance σ2
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varX=E(x-E(x))^2; E(x)=μ
random variable is discrete with N equally likely values:
standard deviation σ – square root of variance
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a measure of spread or scatter in population express in original data
3.2 Important Discrete Distributions
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The Hypergeometric Distribution
o Select a random sample of n items without replacement from a lot of N items of which D
are nonconforming or defects
o x: number of nonconforming items found in the sample
o Hypergeometric probability distribution:
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Example: suppose a lot contains 100 items (N=100), 5 of which do not conform to
requirements (D=5). If items selected without replacement (n=10), then: probability of
finding one or fewer nonconforming items in sample (x=1 or x=0)
 N=100, D=5, n=10, x=1 or x=0
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Binomial Distribution
o Basis in in Bernoulli trails
o A random variable x ~ Bernoulli (p) distribution
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 Value x=1 “success”; p: success probability
Recall: x discrete r.v.
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E(X)=1*p+0*(1-p)=p  μ=E(X)=p (for one trial)
Var(X)=(1-p)2*p+(0-p)2*(1-p)=p(1-p) (for one trial)
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 Mean & variance:
 If X~binomial(n,p), then E(X)=np, VAR(X)=np(1-p)
Poisson Distribution
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Binomial distribution definition
 Notation: random variable x is number of successes out of n Bernoulli trials, with
constant probability of success p on each trial
 Binomial distribution with parameters n>=0 and 0<p<1 is
A typical application Poisson distribution in quality control: number of defects or
nonconformities that occur in a unit of product
Example: x=number of wire-bonding defects occur in a semiconductor device, Poisson
distribution λ=4
 Q: probability that a randomly selected semiconductor device will contain two
or fewer wire-bonding defects
Negative Binomial Distribution
o Consider a sequence of independent trials, each with probability success p, let x denote
trial on which rth success occurs. Then x is a negative binomial random variable with
probability distribution defined as follows.
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Probability of r-1 success in x-1 trial is binomial distribution
probability p success on xth trial multiplying these
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probability gives
Compare: binomial distribution & negative binomial distribution
o Binomial distribution:
 Fix sample size n (# of Bernoulli trials)
 Observe # of success x
o Negative binomial distribution:
 Fix number of success r
 Observe sample size x required to achieve them
 When r=1, negative binomial distribution is “geometric distribution”
Geometric Distribution
o Definition: random variable X number of Bernoulli trial until the first success
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o
Example: medical records
 GGGGB
 G: good
 B: an error
 If probability of finding a bad record is 0.05 (p=0.05), then find probability of
finding a bad record on fifth record examined
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3.3 Important Continuous Distributions
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Normal Distribution
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X~N(μ,σ2)
Cumulative normal distribution
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Definition: probability normal random variable x is less than or equal to some value a,
or:
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Standardization:
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Φ(): cumulative distribution function of standard normal distribution (mean=0,
standard deviation=1)
 Table cumulative standard normal distribution: Appendix Table II
Example3.7: x: time to resolve customer complaints, mean=40, standard deviation =2; Q:
probability that a customer complaint resolved <= 35 hours
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Example3.8: X: diameter of a metal shaft, mean=0.2508 inch, standard deviation 0.0005
inch  X~N(0.2508, 000052), Specifications 0.25±0.0015 inch
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Suppose recenter process mean to 0.2500
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Summary:
Example 3.9: find a particular value of a normal random variable that results in a given
probability
 X~N(10,9), find value of x, say, a, such that P{X>a}=0.05 (Appendix table II)
Lognormal Distribution
o Definition: Let w have a normal distribution mean θ and variance , then x=exp(w) is a
lognormal random variable
Chapter 4 Inferences About Process Quality
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4.1 Statistics and Sampling Distributions
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Statistical inference: draw conclusions about populations (or process) based on sample data
from that system
Random sample: a sample selected so that observations are independent – same probability of
selection
Statistics: any function of observations in sample
o Eg: sample mean, sample variance, sample standard deviation
Sampling distributions
o Statistic – a random variable; probability distribution
o Sampling distribution – probability distribution of a statistics
o Question: let x1, x2, …, xn be a random sample size n from a distribution (population)
with mean μ and variance σ^2. What is the mean (expected value) and variance of
sample mean x-bar?
Three important and useful sampling distributions based on normal distribution
o Chi-square distribution
 If x1, x2, …, xn are normally and independently distributed random variable with
mean 0 and variance 1, then random variable is distributed as chi-square with n
degrees of freedom
o t distribution
 if x is standard normal random variable and if y is a chi-square random variable
with k degrees of freedom, and if x and y are independent
 then random variable:
is distributed as t with k d.f.
o
F distribution
 if w and y are two independent chi-square random variable with u and v degrees
of freedom, then
is distributed as F with u: numerator df and v: denominator df
4.2 Point Estimation of Process Parameters
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Distributions described by their parameters
Parameters unknown, must be estimated
Estimator: define an estimator of an unknown parameter as statistic that corresponds to
parameter
Point estimator: a statistic that produce a single numerical value as “estimate” of parameter
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Example: consider random variable x~N(μ,σ^2)  mean and variance both known
If a random sample of n observations, then sample mean x-bar and sample variance S^2 are
point estimators of population mean μ and population variance
Suppose random variable x inside diameter n=20 bearings. Sample mean x-bar = 1.495; sample
variance S^2 = 0.001
4.3 Statistical Inference for a single sample
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Three components of statistical inference
o Point estimation
o Hypothesis testing
 Null hypothesis vs.. alternative hypothesis
 Statistical hypothesis: a statement about values of parameter of a probability
distribution
 Two-sided alternative hypothesis vs. one-sided alternative hypothesis
o Confidence intervals
 Refers to the probability that a population parameter will fall between pair of
values around mean
 Measure two degree of uncertainty in a sampling method
 Constructed by confidence levels of 95% or 99%
To test a hypothesis
o Take a random sample from population
o Compute an appropriate test statistic
o Either reject or fail to reject the null hypothesis
o “critical region” or “rejection region”: set of values of test statistic leading to rejection of
H0
Two kinds of error when testing hypothesis
o Type I error: if null hypothesis is rejected when it is true
o Type II error: if null hypothesis is not rejected when it is false
In quality control work:
o α: producer’s risk (probability that a good lot will be rejected)
o β: consumer’s risk (probability of accepting a lot of poor quality)
Confidence interval
o An interval estimate of a parameter is interval between two statistics that include true
value of parameter with some probability
o For example, construct an interval estimator of mean μ, must find two statistics L and u
such that P{L <= μ <= u} = 1-α
o Interval L <= μ <= u is called 100(1-α)% confidence interval
o For unknown mean μ, L and u – lower and upper confidence ci units
o 1-α: confidence coefficient
Interpretation of CI:
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If a large number of such intervals are constructed, each resulting from a random
sample, then (1-α)% of these intervals will contain true value of μ
o One-sided lower 100(1-α)% confidence bound on μ  L <= μ
 L: lower confidence bound so that P{L<=μ}=1-α
o One-sided upper 100(1-α)% confidence bound on μ  μ <= U
 U: upper confidence bound so that P{μ<=U}=1-α
Confidence interval on mean with variance known
o Consider random variable x, with unknown mean μ and known variance. Suppose a
random sample of n observations x1, x2, … ,xn.
o Then the 100(1-α)% two-sided CI on μ is:
o
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4.3.2 The Use of P-Values for Hypothesis Testing
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P-value is the probability that test statistic will take on a value that is at least as extreme as
observed value of statistic when null hypothesis is true
The p-value is smallest level of significance that would lead to rejection of null hypothesis
Normal distribution, compute p-value
4.3.3 Inference on Mean of a Normal Distribution, Variance Unknown
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Hypothesis testing: x normal random variable with unknown mean and unknown variance
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