STAT 528
HOMEWORK #5
Due: Friday, October 2, 2015.
13 points
1. (3.5pt) Exercise 7.37, page 338 (7.49, page 349). Be sure to include a u-chart.
2. (3.5pt) Exercise 7.40, page 339 (7.52, page 349). Change the number of nonconformities
from 5 to 6 for Deck Number 2425. Include a c-chart. You can also assume that an
assignable cause was found for any out-of-control signal, and that data point can be
removed, and the c-chart can be revised.
3. (13pt) I will email the following SAS code to you. This file contains a data set of 200
breaking strength measurements. These measurements were taken from an in-control
process and are listed in increasing order. The target is 200 pounds with LSL = 185 and
U SL = 215 pounds.
(a) Find the ‘best’ fitting normal, gamma, weibull, beta, and lognormal distributions
that model the data. Provide the estimates of the parameters for each distribution.
For the gamma, Weibull, and lognormal distributions this will involve choosing a
lower threshold θ. To fit a beta distribution, you need to specify a lower threshold θ
and a constant σ to add to θ that will determine the upper threshold θ + σ. For the
normal distribution, no threshold value is necessary.
(b) Which fitted distribution do you recommend? Why? Include a cdf plot with the
superimposed fitted cdf and a histogram with the superimposed fitted density.
(c) Find Cbp , Cbpk and Cbpm for your fitted distribution. Interpret these three values.
OPTIONS NODATE NONUMBER
DATA IN; INPUT STRENGTH
195.1 195.3 195.7 196.0
196.9 197.0 197.3 197.3
197.6 197.6 197.7 197.7
198.1 198.1 198.1 198.1
198.8 198.9 198.9 198.9
199.5 199.6 199.6 199.6
200.2 200.2 200.4 200.5
201.0 201.0 201.0 201.2
201.6 201.6 201.6 201.8
202.3 202.3 202.4 202.5
203.0 203.1 203.1 203.2
204.3 204.3 204.3 204.5
205.1 205.4 205.6 205.7
206.2 206.2 206.4 206.4
207.7 207.7 207.8 207.9
210.1 210.2 210.3 211.3
213.3 213.4 213.5 213.5
;
symbol v=dot width=3;
LS=72;
@@; LINES;
196.0 196.0
197.3 197.4
197.8 197.8
198.2 198.3
198.9 198.9
199.6 199.7
200.6 200.6
201.2 201.3
201.8 201.8
202.5 202.6
203.6 203.8
204.6 204.7
205.7 205.8
206.6 206.6
208.0 208.0
211.4 211.4
213.5 214.8
196.2
197.4
197.8
198.4
199.0
199.8
200.6
201.3
201.9
202.7
203.8
204.9
205.8
206.6
208.1
211.5
215.3
196.4
197.5
197.9
198.6
199.1
199.9
200.7
201.3
201.9
202.7
203.9
204.9
205.9
206.7
208.3
211.7
216.4
196.4
197.5
197.9
198.7
199.1
199.9
200.7
201.4
201.9
202.8
203.9
205.0
205.9
207.0
208.4
212.5
PROC CAPABILITY DATA=IN ;
VAR STRENGTH;
SPEC LSL=185 USL=215 TARGET=200 ;
CDFPLOT
STRENGTH / NORMAL(INDICES);
HISTOGRAM STRENGTH / NORMAL(INDICES)
MIDPOINTS=185 TO 215 BY 3;
INSET MEAN (5.3) STD=’Std Dev’ (5.3) /
HEADER = ’Normal Fit’ POS = NE;
TITLE ’PROCESS CAPABILITY COMPARISON -- NORMAL FIT’;
196.5
197.5
197.9
198.7
199.3
200.0
200.8
201.5
202.0
202.9
203.9
205.0
206.0
207.1
208.5
212.6
196.6
197.5
198.0
198.7
199.3
200.1
200.9
201.5
202.0
203.0
204.0
205.0
206.1
207.4
208.8
212.8
196.7
197.6
198.0
198.7
199.5
200.2
200.9
201.5
202.2
203.0
204.1
205.0
206.1
207.5
209.1
212.9
/*
*********************************************************;
*** For a nonstandard gamma, THETA = lower threshold ***;
*********************************************************;
PROC CAPABILITY DATA=IN;
VAR STRENGTH;
SPEC LSL=185 USL=215 TARGET=200 ;
CDFPLOT
STRENGTH / GAMMA(THETA=xx INDICES);
HISTOGRAM STRENGTH / GAMMA(THETA=xx INDICES)
MIDPOINTS=185 TO 215 BY 3;
INSET MEAN (5.3) STD=’Std Dev’ (5.3)
SKEWNESS (5.3) KURTOSIS (5.3) /
HEADER = ’Gamma Fit’ POS = NE;
TITLE ’PROCESS CAPABILITY COMPARISON -- GAMMA FIT’;
*/
/*
***********************************************************;
*** For a nonstandard Weibull, THETA = lower threshold ***;
***********************************************************;
PROC CAPABILITY DATA=IN;
VAR STRENGTH;
SPEC LSL=185 USL=215 TARGET=200 ;
CDFPLOT
STRENGTH / WEIBULL(THETA=xx INDICES);
HISTOGRAM STRENGTH / WEIBULL(THETA=xx INDICES)
MIDPOINTS=185 TO 215 BY 3;
INSET MEAN (5.3) STD=’Std Dev’ (5.3) /
HEADER = ’Weibull Fit’ POS = NE;
TITLE ’PROCESS CAPABILITY COMPARISON -- WEIBULL FIT’;
*/
/*
**********************************************************************;
*** Two threshold values must be specified for a nonstandard beta ***;
***
THETA = lower threshold , THETA+SIGMA = Upper threshold
***;
**********************************************************************;
PROC CAPABILITY DATA=IN;
VAR STRENGTH;
SPEC LSL=185 USL=215 TARGET=200 ;
CDFPLOT
STRENGTH / BETA(THETA=xx SIGMA=xx INDICES);
HISTOGRAM STRENGTH / BETA(THETA=xx SIGMA=xx INDICES)
MIDPOINTS=185 TO 215 BY 3;
INSET MEAN (5.3) STD=’Std Dev’ (5.3) /
HEADER = ’Beta Fit’ POS = NE;
TITLE ’PROCESS CAPABILITY COMPARISON -- BETA FIT’;
*/
/*
*************************************************************;
*** For a nonstandard lognormal, THETA = lower threshold ***;
*************************************************************;
PROC CAPABILITY DATA=IN;
VAR STRENGTH;
SPEC LSL=185 USL=215 TARGET=200 ;
CDFPLOT
STRENGTH / LNORM(THETA=xx INDICES);
HISTOGRAM STRENGTH / LNORM(THETA=xx INDICES)
MIDPOINTS=185 TO 215 BY 3;
INSET MEAN (5.3) STD=’Std Dev’ (5.3) /
HEADER = ’Lognormal Fit’ POS = NE;
TITLE ’PROCESS CAPABILITY COMPARISON -- LOGNORMAL FIT’;
*/
RUN;