Elementary Statistical Quality Control 2nd Edition John T. Burr - Solutions

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If a signifi cant correlation is obtained and the causation inferred, could there be a third factor, not measured, which is jointly correlated with the other two, and so accounts for their association?
Remember correlation does not necessarily imply causation.
If a scatter plot is given to support a linear regression, is the variability of the points about the line roughly the same over the range of the independent variable? If not, then perhaps some transformation of the variables is necessary before computing the regression line.
If predictions are given, are any made outside the range of the observed values of the independent variable?
A survey was conducted in 295 people asking about arthritic pain on a visual analogue scale (can be treated as continuous) in 295 people. Sex was coded 1 = male 0 = female. Medication was coded 1 = on medication 0 = not on medication.The output from a multiple linear regression computer program is
A survey was conducted and asked ‘do you consider your health to be poor?’ This was coded 1 for ‘yes’ and 0 for ‘no’. The effect of age(in decades) and sex on the outcome was examined using logistic regression.(i) Deduce the values of A B and C.(ii) Are either age or sex signifi cant
Is the nature of the censoring specifi ed? As far as possible check that the censoring is non-informative.
Check that the total number of critical events is reported, as well as subjects and person-time of follow-up, with some measure of variability such as a range for the latter. In trials, are numbers of censored observations given by treatment group?
Is the estimated survival rate at a conventional time point, by group, with confi dence intervals given?
Are the Kaplan–Meier survival curves by group displayed?
Are the numbers at risk reported at regular time intervals?
Is the regression model used specifi ed?
Is the proportional hazards assumption reasonable and has it been validated?
If a multivariable Cox model is used, are the corresponding HRs and CIs reported for all the variables included?
Are the conclusions drawn critically dependent on the statistical assumptions made?
Is the computer program used for the analysis indicated?
When a new instrument has been used, has it been tested for reliability and validity?
When two methods of measurement have been compared, has correlation been used to assess whether one instrument can be used instead of another?If so are there systematic biases?
When Cohen’s kappa is compared, are the marginal distributions similar?If not, then be careful about making comparisons.
Collecting a urine sample in young non-toilet trained children can be challenging. The standard method is a urine collection bag device, but this is uncomfortable and can be contaminated. Farrell et al (2002) compare the bag, with the use of an absorbent sanitary pad. They studied 20 children and
Bland and Altman (1997) describe the mini-HAQ that measures impairment in patients with cervical myelopathy. There are 10 items relating to activities of daily living: stand, get out of bed, cut meat, hold cup, walk, climb stairs, wash, use toilet, open a jar and enter/leave a car. The standard
Consistency. Have other investigators and other studies in different populations led to similar conclusions?
Plausibility. Are the results biologically plausible? For example, if a risk factor is associated with cancer, are there known carcinogens in the risk factor?
Dose–response. Are subjects with a heavy exposure to the risk factor at greater risk of disease than those with only slight exposure?
In a cohort study, have a large percentage of the cohort been followed up, and have those lost to follow-up been described by their measurements at the start of the study?
How has the cohort been selected? Is the method of selection likely to infl uence the variables that are measured?
In a case–control study, are the cases likely to be typical of people with the disease? If the cases are not typical, how generalisable are the results likely to be?
In a matched case–control study, has allowance been made for matching in the analysis?
In any observational study, what are the possible biases? How have they been minimised and have the Bradford Hill criteria been considered?
What type of study is being described in each of the following situations?(a) All female patients over the age of 45 on a general practitioner’s list were sent a questionnaire asking whether they had had a cervical smear in the last year.(b) A group of male patients who had had a myocardial
Many observational studies have shown that women taking hormone replacement therapy (HRT) are at lower risk of heart disease. What biases might be involved in these conclusions?
Are the trial objectives clearly formulated?
Are the diagnostic criteria for entry to the trial clear?
Is there a reliable supply of patients?
Are the treatments (or interventions) well defi ned?
Is the method and reason for randomisation well understood?
Is the treatment planned to commence immediately following randomisation?
Is the maximum degree of blindness being used?
Are the outcome measures appropriate and clearly defi ned?
How has the study size been justifi ed?
What arrangements have been made for collecting, recording and analysis of the data?
Has appropriate follow-up of the patients been organised?
Are important prognostic variables recorded?
Are side-effects of treatment anticipated?
Are many patient drop-outs anticipated?
Are the statistical procedures used adequately described or referenced?
Are the statistical procedures appropriate?
Have the potential prognostic variables been adequately considered?
Is the statistical presentation satisfactory?
Are any graphs clear and the axes appropriately labelled?
Are confi dence intervals given for the main results?
Are the conclusions drawn from the statistical analysis justifi ed?
Was the analysis conducted by ‘intention to treat’ or ‘per protocol’?
Is the actual size of the treatment effect reported, and the associated confi dence interval reported?
Does the Abstract correctly report what was found in the paper?
Have the CONSORT suggestions been followed?
Suppose we wish to estimate the prevalence of nurses dissatisfi ed with their job. We think it will be about 20%, and we would like to estimate this to within 5%. How many subjects do we need?
Hippisley-Cox et al (2003) described a survey of the results of various statins on serum cholesterol in general practice. They found that the serum cholesterol for 554 patients on atorvastatin to be 4.99 mmol/l (95% CI 4.90 to 5.09). Suppose a new statin came on the market and we wished to design a
Suppose we are planning an exercise trial in 50–74-year-old men, to see if a daily exercise regime for a year will lead to improved quality of life compared to a control group. We know from published data on the SF-36 quality of life measure that at this age men have a mean score on the Physical
Suppose we are planning an exercise trial in 50–74-year-old men identifi ed as at high risk of a heart attack, to see if a daily exercise regime for a year will lead to a reduction in the number of heart attacks. One group will be given the daily exercise regime and the other control group will
Are the distributional assumptions underlying parametric tests such as the t-test satisfi ed? Is there any way of fi nding out?
If a correlation coeffi cient is tested for signifi cance is the null hypothesis of zero correlation a sensible one?
Is the study a repeated measures type? If so, beware! Read the paper by Matthews et al (1990) for advice on handling these types of data.
Are the results clinically signifi cant as well as statistically signifi cant? If the results are statistically not signifi cant, is equivalence between groups being claimed? If the result is statistically signifi cant, what is the size of the effect? Is there a confi dence interval quoted?
Have a large number of tests been carried out that have not been reported?Were the hypotheses generated by an exploration of the data set, and then confi rmed using the same data set?
Have the subjects been selected because of a high or low value of a particular variable, and this variable subsequently remeasured? If so beware!
A police authority installed speed cameras at accident black spots. In the following year they noted that the number of accidents at these sites had fallen from the year before the cameras were installed. Can one thus conclude that speed cameras are successful in reducing accidents?
A group of children were identifi ed as slow learners. They were then randomly allocated to take either a cod liver oil capsule or a dummy placebo in a double blind trial. After 6 months the group allocated cod-liver oil appeared to have improved in terms of reading ability, relative to the dummy
8.1. Obtain 30 stratified samples, each of three x values, consisting of the total points in a throw of two dice, then three dice and finally four dice. Tabulate the three x values always in the same order as in Table 8.1. (a) Analyze the stratified samples as in rows: using the usual control chart
8.2. Simulate a three-spindle automatic being run for production at specification limits of 5 (coded). Use 30 samples of n¼3 each consisting of one from each of populations A, B and F of Table 7.3. Always take in the same order as in Table 8.1. (a) Analyze the stratified samples in rows using the
8.3. This problem involves measurements on the outside diameter at the base of the stem of an exhaust-valve bridge.Specifications were 1.1550–1.1560 in. All pieces from the machining operation were measured and samples of n¼5 in succession were formed. Three vendors were supplying forgings.There
8.4. The data shown below constituted the original run preceding that of Table 8.3. As in that table, specifications were 0.1235 to 0.1250 in. with data in 0.0001 in. units below 0.1250 in. and n¼5.Time July 27 8:00 8:30 9:00 9:30 10:00 11:00 12:30 x-bar 8.0 7.4 6.8 5.4 5.6 4.0 3.4 R 2 3 2
8.5. The following analyses, x, give the carbon content of 20 consecutive heats of 1045 steel. Three heats are made in an open hearth furnace per day. Check the control by an x chart(with 2s limits) and a moving range chart. Specified limits are 0.45% to 0.50%. The following are given in 0.001%
8.6. A steel plant was producing hot-rolled bars which were shipped to a customer who cold drew them. After cold drawing, the bars were inspected and certain ones rejected for defects purporting to originate in the steel plant. Rejections were totaled for each month and expressed as a percentage of
8.7. Give two examples from your own experience of data on fraction nonconforming which cannot appropriately be analyzed by a p chart. State why not in each case.
8.8. Find the average run lengths in terms of individual pieces, when using an upper control limit of 17 (17 nonconformities are out of control) for the total number of nonconformities on three units: (a) when co for one piece is 3 and (b) when co for one piece is 6.
8.9. Suppose that on 100 units of equipment, P P c1¼1, c2¼3, P c3¼32, P c4¼350 with corresponding weights w1¼100, w2¼50, w3¼10, w4¼1. Standard values for co are 0.001, 0.005, 0.3, and 3, respectively for 1, 2, 3 and 4. Find U and compare with control lines.
9.1. For the example of single sampling given in Section 9.5 check the results given for Pa, AOQ, and ATI at po¼0.02 and at po¼0.06. Find p95 and p10 for the plan by reversing the use of Table B, that is, finding npo when P(3 or less) is 0.95 and 0.10.
9.2. For the example of double sampling given in Section 9.6, check the results given for Pa, AOQ, ASN, and ATI at po¼0.02 and at po¼0.06.
9.3. Sketch the OC, AOQ, ASN, and ATI curves labeling units on the axes, for the following two acceptance sampling plans for lots of 1000: (a) n¼80, Ac¼2, Re¼3 and (b)n1¼n2¼50, Ac1¼0, Re1¼3, Ac2¼3, Re2¼4. Points for po¼0.01, 0.02, . . . , 0.06 should be enough.
9.4. Sketch the OC, AOQ, ASN, and ATI curves, labeling units on the axes for the single sampling plan: N¼2000, n¼100, Ac¼2, Re¼3. Also find the AOQL, and p95 and p10.Use points for po¼0.01, 0.02, . . . , 0.05 initially.
9.5. Sketch the OC, AOQ, ASN, and ATI curves, labeling units on the axes for the double sampling plan: N¼2000, n1¼50, n2¼100, Ac1¼1, Re1¼4, Ac2¼3, Re2¼4. Use points for po¼0.01, 0.02, . . . , 0.06 which may be enough.
9.6. Find a single sampling plan (using Section 9.8) having p95¼p1¼0.01 and p10¼p2¼0.05. Similarly, find one for p1¼0.002, p2¼0.01.
9.7. Find a single sampling plan (using Section 9.8) having p95¼p1¼0.01 and p10¼p2¼0.03. Also find one for p1¼0.001, p2¼0.003.
10.1. Find the following plans from the ANSI=ASQ Z1.4:Lot size Inspection level S, D, or M N, T, or R Nonconforming or nonconformities AQL(%)(a) 500 II Single Norma1 Nonconforming 0.40(b) 500 II Single Tightened Nonconforming 0.40(c) 500 II Single Reduced Nonconforming 0.40(d) 500 I Single Normal
10.2. Find single sampling plans in the ABC Standard for lot size N¼500, inspection level II, with AQL¼2.5 (nonconforming)for (a) normal, (b) tightened, and (c) reduced.Draw the three OC curves on the same axes, labeling the scales. Comment on the curves.
10.3. A Dodge–Romig sampling plan for an AOQL of 1%and lots of 3000 calls for n1¼80, n2¼170, Ac1¼0, Re1¼5, Ac2¼4, Re2¼5. Find Pa when po¼0.04. Is this compatible with the listed p10¼0.036?
10.4. A Dodge–Romig single sampling plan for lots of N¼3000 and p-bar¼0.21–0.40% is n¼80, Ac¼1. It is for an AOQL of 1.0%. Draw the OC and AOQ curves and check this AOQL.
10.5. Find the CSP-1 continuous sampling plan for an AOQL of 1.0% with f¼0.125. Explain the meaning of i andf. What is the corresponding CSP-2 plan? Again explain i and f.
10.6. In what sense is the OC curve for the ChSP-1 chain sampling plan n¼10, Ac¼0, i¼2 more favorable than that for the ordinary sampling plan n¼10, Ac¼0?
10.7. Explain in simple terms the operation of a SkSP-l skip sampling plan having AOQL¼2% and f¼1=2. What advantages and disadvantages do it have?
11.1. The time of blow for fuses under a marginal circuit condition is subject to a maximum specification limit of 150 sec¼U. Experience has shown that sx¼25 sec. Regarding m1¼100 sec as the AQL and m2¼125 sec as the RQL, and setting respective risks of a¼0.10 and b¼0.10, find an appropriate
11.2. The ‘‘Scott value’’ for material for a battery is subject to a maximum specification of 26 and sx is known from experience to be 0.56. Assume normality. Set up a sampling plan so that if m¼m1¼24.5, the probability of rejection is 0.05, whereas if m¼m1¼25.0, the probability of
11.3. For the weight of contents of a package of a food product, the minimum specification for weights is 500 g. From past experience sx, is known to be 0.6 g. Taking m¼m1¼501 g as the RQL, and m¼m2¼502 g as the AQL, and using respective risks a and b both at 0.05, find a sampling plan. Assume
11.4. For the length of a spring under a compressive force of 50 kg, there is set minimum limit of 10.0 cm. Assume that sx¼0.2 cm, and that the distribution of the x’s is normal.Set m¼m1¼10.4cm as the RQL and m¼m2¼10.6 cm as the AQL, and take risks of b¼a¼0.05. Find a sampling plan for the
11.5. For doubled thickness of rubber gaskets for metal tops for food jars, sx was known to be 0.002 in., and the distribution of x’s to be normal. Consider m¼0.106 in.¼AQL and m¼0.103 in. or 0.109 in. as RQL’s. Set a¼b¼0.01. (Actually, if m is at either RQL, the rubber is used in another
11.6. A small part has diameter specifications of 0.1100 and 0.1102 in., each lot being produced under a single setup.Experience has shown a normal distribution of diameters with sx, ¼0.00003 in. Taking m¼mo¼0.1101 in.¼AQL and m¼mo mo¼0.00003 in. as the RQL’s and setting a and b risks at
11.7. Devise a test of variability for lots having s1¼1.715, s2¼3.43, a¼b¼0.10, assuming normality. Sketch the OC curve Pa vs. s, labeling axes. Test once each on distributions A and C of Table 7.3, by experimentation.
11.8. Muzzle velocity of target ammunition is to be tested for variability. Take s1¼5 ft=sec, and use risks a¼b¼0.05.Set up a sampling plan to make the test. Sketch the OC curve, Pa vs. s, labeling axes.
11.9. A new gauge is being considered. If the standard deviation, s1, on homogeneous material is 0.00002 in., the gauge should be approved, whereas if at s2¼0.00003 in., it should be rejected. Use risks of a¼b¼0.10 and find a sampling plan.

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Elementary Statistical Quality Control 2nd Edition John T. Burr - Solutions

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