Reliability plotting is a method to estimate the percentage of a population that is within specification limits using a probability plot of sample data. It involves: 1) creating a cumulative probability plot of the data, 2) transforming the plot axes to straighten the curve, 3) extrapolating the straight line to the specification limit, 4) calculating the confidence limit of the extrapolated point, 5) back-transforming to find the percentage out of specification, and 6) subtracting from 100% to obtain the reliability percentage.
Cumulative Frequency Table,Cumulative Frequency Table with Example,Ogive Curve,Two Types of Ogive Curve,Less than Ogive with Example,Greater than Ogive with Example.
Cumulative Frequency Table,Cumulative Frequency Table with Example,Ogive Curve,Two Types of Ogive Curve,Less than Ogive with Example,Greater than Ogive with Example.
Types of Graphs
(i) Graph of time series or Historigram
(ii) Histogram
(iii) Frequency polygon
(iv) Frequency curve
(v) Cumulative Frequency polygon or Ogive
Historigram
A Historigram is constructed by taking time along X-axis and the value of the variable along Y-axis. Points are plotted and are then connected by straight line segments to get the Historigram.
Types of Graphs
(i) Graph of time series or Historigram
(ii) Histogram
(iii) Frequency polygon
(iv) Frequency curve
(v) Cumulative Frequency polygon or Ogive
Historigram
A Historigram is constructed by taking time along X-axis and the value of the variable along Y-axis. Points are plotted and are then connected by straight line segments to get the Historigram.
FSE 200AdkinsPage 1 of 10Simple Linear Regression Corr.docxbudbarber38650
FSE 200
Adkins Page 1 of 10
Simple Linear Regression
Correlation only measures the strength and direction of the linear relationship between two quantitative variables. If the relationship is linear, then we would like to try to model that relationship with the equation of a line. We will use a regression line to describe the relationship between an explanatory variable and a response variable.
A regression line is a straight line that describes how a response variable y changes as an explanatory variable x changes. We often use a regression line to predict the value of y for a given value of x.
Ex. It has been suggested that there is a relationship between sleep deprivation of employees and the ability to complete simple tasks. To evaluate this hypothesis, 12 people were asked to solve simple tasks after having been without sleep for 15, 18, 21, and 24 hours. The sample data are shown below.
Subject
Hours without sleep, x
Tasks completed, y
1
15
13
2
15
9
3
15
15
4
18
8
5
18
12
6
18
10
7
21
5
8
21
8
9
21
7
10
24
3
11
24
5
12
24
4
Draw a scatterplot and describe the relationship. Lay a straight-edge on top of the plot and move it around until you find what you think might be a “line of best fit.” Then try to predict the number of tasks completed for someone having been without sleep 16 hours.
Was your line the same as that of the classmate sitting next to you? Probably not. We need a method that we can use to find the “best” regression line to use for prediction. The method we will use is called least-squares. No line will pass exactly through all the points in the scatterplot. When we use the line to predict a y for a given x value, if there is a data point with that same x value, we can compute the error (residual):
Our goal is going to be to make the vertical distances from the line as small as possible. The most commonly used method for doing this is the least-squares method.
The least-squares regression line of y on x is the line that makes the sum of the squares of the vertical distances of the data points from the line as small as possible.
Equation of the Least-Squares Regression Line
· Least-Squares Regression Line:
· Slope of the Regression Line:
· Intercept of the Regression Line:
Generally, regression is performed using statistical software. Clearly, given the appropriate information, the above formulas are simple to use.
Once we have the regression line, how do we interpret it, and what can we do with it?
The slope of a regression line is the rate of change, that amount of change in when x increases by 1.
The intercept of the regression line is the value of when x = 0. It is statistically meaningful only when x can take on values that are close to zero.
To make a prediction, just substitute an x-value into the equation and find .
To plot the line on a scatterplot, just find a couple of points on the regression line, one near each end of the range of x in the data. Plot the points and connect them with a line. .
Pampers CaseIn an increasingly competitive diaper market, P&G’.docxbunyansaturnina
Pampers Case
In an increasingly competitive diaper market, P&G’s marketing department wanted to formulate new approaches to the construction and marketing of Pampers to position them effectively against Hugggies without cannibalizing Luvs. They surveyed 300 mothers of infants. Each was given a randomly selected brand of diaper (either Pampers, Luvs, or Huggies) and asked to rate that diaper on nine attributes and to give her overall preference for the brand. Preference was obtained on a 7-point Likert scale (1=not at all preferred, 7=greatly preferred). Diaper ratings on the nine attributes were also obtained on 7-point Likert scale (1=very unfavorable, 7=very favorable). The study was designed so that each of the three brands appeared 100 times. The goal of the study was to learn which attributes of diapers were most important in influencing purchase preference (Y). The nine attributes used in study were:
Variable
Attribute
Marketing options
X1
count per box
Desire large counts per box?
X2
price
Pay a premium price?
X3
value
Promote high value
X4
skin care
Offer high degree of skin care
X5
style
Prints/color vs. plain diapers
X6
absorbency
Regular vs. superabsorbency
X7
leakage
Narrow/tapered vs. regular crotch
X8
comfort/size
Extra padding and form-fitting gathers
X9
taping
Re-sealable tape vs. regular tape
Question (will be discussed in week 8):
If you don’t have SPSS software at home, you may be able to download a trial version (good for 21 days) from spss.com(software(statistics family(PASW statistics 17.0(click “free trial” and download.
1. Run a regression analysis for brand preference that includes all independent variables in the model, and describe how meaningful the model is. Interpret the results for management.
6. Correlation and Regression
*
The mean, or average value, is the most commonly used measure of central tendency. The mean, ,is given by
Where,
Xi = Observed values of the variable X
n = Number of observations (sample size)
The mode is the value that occurs most frequently. It represents the highest peak of the distribution. The mode is a good measure of location when the variable is inherently categorical or has otherwise been grouped into categories.
Statistics Associated with Frequency Distribution Measures of Location
X
=
X
i
/
n
S
i
=
1
n
X
*
The median of a sample is the middle value when the data are arranged in ascending or descending order.
http://www.city-data.com/
Statistics Associated with Frequency Distribution Measures of Location
*
Skewness. The tendency of the deviations from the mean to be larger in one direction than in the other. It can be thought of as the tendency for one tail of the distribution to be heavier than the other.
Kurtosis is a measure of the relative peakedness or flatness of the curve defined by the frequency distribution. The kurtosis of a normal distribution is zero. If.
CASE 02: SAIGON COOPMART
Logistics & Supply Chain plays an important role, if needed to say a critical factor for the success of Saigon Coopmart. Most of supermarkets over the world follow the identical model in which a warehouse is placed next to supermarket for stocks storage; and the size of warehouse is more or less equal to size of supermarket. However, due to harsh competition, and weak finance, Saigon Coopmart decided to follow a different model with very small size warehouse. This allows Saigon Coopmart to place more supermarkets; but in exchange, stocks only enough for a day, or maximum two compared to ordinary model in which a warehouse can store enough stocks for a week or more. As a consequence, Saigon Coopmart has to ship much more frequency to its supermarkets than its competitors such as Big C.
Option pricing under quantum theory of securities price formation - with copy...
Reliability Plotting Explained
1. RELIABILITY PLOTTING
Reliability Plotting is a method for determining the shape of a distribution and using that knowledge to
estimate what percentage of that population is "in specification" (that percentage is commonly
referred to as the "% Reliability"). The distribution's shape is identified by means of a "Probability Plot"
of the measurement ("variables") data derived from a random or representative sample taken from
the population.
That entire process is summarily described in the following 6 sequential steps, for a one-sided lower
specification limit:
1. The sample data values are charted onto a "cumulative probability plot", with the X-axis values
being the raw data points arranged in numerical order with the lowest values on the left-hand
side of the axis, and with the Y-axis values being the corresponding cumulative percentages.
For the process being described here, the decimal % cumulative value that corresponds to
each X-axis value is approximately the ratio of the data-point's rank divided by "N" (which is
the total number of data points); e.g., if N=15, the X-axis's lowest data-point's % cumulative
equals approximately 1/15, the median value on the X-axis equals approximately 8/15, and the
largest value on the X-axis equals approximately 15/15. The resulting plot typically looks like
an "S" shaped curve, with the Y-axis decimal values ranging from near 0.00 to near 1.00
(which is the equivalent of a range from near 0% to near 100%).
2. "Transformations" are chosen for the Y-axis values and/or the X-axis values in attempts to
straighten-out that S-shaped curve. Common transformation are Log, square-root, or inverse,
as well as others such as Z-table, t-table, or Chi-square table values; most of those
transformations have been in use since reliability plotting was first invented, in the mid-20th
century. In most cases, the best straight line is the one that generates the highest Correlation
Coefficient, as determined by the classic method of Linear Regression.
3. After the combination of X-and-Y axis transformations is found that provides the best straight
line, that line is extrapolated to the point on the X-axis that corresponds to the QC or
Verification/Validation pass/fail criterion (a.k.a., "spec limit").
4. The Y-axis value that corresponds to that extrapolated X-axis value then has its 1-sided upper
confidence limit calculated (typically, a 95% confidence limit is chosen), using the classic
method that is appropriate for the mean of a point on a Linear Regression line.
5. That confidence limit is then back-transformed. For example, if the Y-axis had been
transformed by taking the Log(base 10) of every % cumulative value, then the back-
transformation is achieved by taking the anti-log of the confidence limit (that is, using the
confidence limit as a power on a base of 10). The resulting value represents the decimal % of
the population that is out-of-specification.
6. That %-out-of-specification value is then subtracted from 100%, to yield the desired result,
namely the % Reliability. It can then be claimed that, at the confidence level used in step 4
above, the % of the population that meets specification is no worse than the calculated %
Reliability.
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v4 Copyright 2012, by John Zorich (www.johnzorich.com, Zorich Consulting & Training)