-SQA-SCOTTISH QUALIFICATIONS AUTHORITY HIGHER NATIONAL UNIT SPECIFICATION GENERAL INFORMATION

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1 -SQA-SCOTTISH QUALIFICATIONS AUTHORITY HIGHER NATIONAL UNIT SPECIFICATION GENERAL INFORMATION -Unit number Unit title- INTRODUCTION TO STATISTICS FOR QUALITY ANALYSIS -Superclass category- -Date of publication- (month and year) -Originating centre for unit- RB JUNE 1999 SQA DESCRIPTION- GENERAL COMPETENCE FOR UNIT: Using the elementary algebraic, statistical techniques and sampling methods required in the study of quality management. OUTCOMES: 1. apply algebraic methods in problem solving; 2. use descriptive statistics; 3. select suitable methods of data collection. CREDIT VALUE: 1 HN Credit ACCESS STATEMENT: Access is at the discretion of the centre. However, it would be beneficial if the candidate possessed general competence in algebraic manipulation. This may be evidenced by the possession of a Standard Grade in Mathematics at Credit level, by Core Maths 4, or by an equivalent level of experience

2 Additional copies of this unit can be obtained from: The Committee and Administration Unit, SQA, Hanover House, 24 Douglas Street, Glasgow G2 7NQ, (Tel: ). At the time of publication the cost is 1.50 per unit (minimum order 5.00). 2

3 HIGHER NATIONAL UNIT SPECIFICATION STATEMENT OF STANDARDS Unit number: Unit title: INTRODUCTION TO STATISTICS FOR QUALITY ANALYSIS Acceptable performance in this Unit will be the satisfactory achievement of the standards set out in this part of the specification. All sections of the statement of standards are mandatory and cannot be altered without reference to SQA. OUTCOME 1. APPLY ALGEBRAIC METHODS IN PROBLEM SOLVING PERFORMANCE CRITERIA (a) (b) (c) (d) The rounding of numbers to a given specification is correct. The numerical evaluation of expressions is correct. The solution of equations is correct. The changing of the subject of formulae is correct. RANGE STATEMENT Rounding specification: significant figures; decimal places; specified unit. Expressions: linear; indicial; exponential; logarithmic. EVIDENCE REQUIREMENTS Four correct solutions are required for performance criterion (a), between them covering the full range of rounding specifications. Two correct solutions are required for each of performance criteria (b), (c) and (d), between them covering the full range of expressions. OUTCOME 2. USE DESCRIPTIVE STATISTICS PERFORMANCE CRITERIA (a) (b) The construction of diagrams is correct for a given context. The calculation of measures of central tendency and spread appropriate to a given context is correct. 3

4 RANGE STATEMENT The range statement for this outcome is fully expressed within the performance criteria. EVIDENCE REQUIREMENTS Recorded evidence is required which indicates that the candidate can fulfil all of the performance criteria. Two different types of diagram are required for performance criteria (a). Two different pairs of measures of central tendency and spread are required for performance criterion (b) which must include mean and standard deviation OUTCOME 3. SELECT SUITABLE METHODS OF DATA COLLECTION PERFORMANCE CRITERIA (a) (b) (c) The choice of data collection method is correct for a given context. The use of British Standards to select a sample size and a sampling method is correct. The selection of a sample using random number tables is correct. RANGE STATEMENT The range statement for this outcome is fully expressed within the performance criteria. EVIDENCE REQUIREMENTS Evidence of the candidate s ability to select suitable methods of data collection on at least two occasions for PC (a), to use at least two different British Standards for PC (b) and to use random number tables at least once for PC (c). MERIT STATEMENT: To gain a pass in this unit, a candidate must meet the standards set out in the outcomes, performance criteria, range statements and evidence requirements. To achieve a merit in this unit, a candidate must demonstrate a superior or more sophisticated level of performance. In this unit this might be shown in the following ways: consistently high level of accuracy; outstanding skills of analysis; consistently logical presentation of work. 4

5 Evidence which satisfies the criteria for merit may be generated by either: solving the problem to a level beyond that defined for a pass or where this is not possible, including in the assessment a further section which would allow the candidate to demonstrate skills which satisfy the criteria for merit. ASSESSMENT In order to achieve this unit, candidates are required to present sufficient evidence that they have met all the performance criteria for each outcome within the range specified. Details of these requirements are given for each outcome. The assessment instruments used should follow the general guidance offered by the Scottish Qualifications Authority (SQA) assessment model and an integrative approach to assessment is encouraged. (See references at the end of support notes). Accurate records should be made of the assessment instruments used showing how evidence is generated for each outcome and giving marking schemes and/or checklists, etc. Records of candidates achievements should be kept. These records will be available for external verification. SPECIAL NEEDS Proposals to modify outcomes, range statements or agreed assessment arrangements should be discussed in the first place with the external verifier. Copyright SQA 1999 Please note that this publication may be reproduced in whole or in part for educational purposes provided that: (i) (ii) no profit is derived from the reproduction; if reproduced in part, the source is acknowledged. 5

6 HIGHER NATIONAL UNIT SPECIFICATION SUPPORT NOTES Unit number: Unit title: INTRODUCTION TO STATISTICS FOR QUALITY ANALYSIS SUPPORT NOTES: This part of the unit specification is offered as guidance. None of the sections of the support notes is mandatory. NOTIONAL DESIGN LENGTH: SQA allocates a notional design length to a unit on the basis of time estimated for achievement of the stated standards by a candidate whose starting point is as described in the access statement. The notional design length for this unit is 40 hours. The use of notional design length for programme design and timetabling is advisory only. PURPOSE This unit can be taken as a free standing single credit unit and can be used as part of the Higher National award programme. It is a mandatory unit in the HND Quality. Further information can be obtained from SQA. CONTENT/CONTEXT The use of a scientific calculator with a statistical facility for direct calculation of mean and standard deviation is imperative. The use of graphics calculators and computer packages with graphical and statistical facilities, is strongly encouraged. All assessments should conform to the relevant standards laid down by the British Standards Institute. Currently these include: BS 2846 Guide to statistical interpretation of data Part 1 Routine analysis of data BS ISO Statistics - Vocabulary and Symbols BS 1957 Presentation of Numerical Values (summarised in BS 2846 Part 1, Appendix A) There are many further British standards which give sampling schemes for different substances (refer to the British Standards index volume under Sampling Methods ). All British Standards are now available on CD-ROM. 6

7 Corresponding to outcomes: 1. Use of a calculator. Scientific notation. Evaluation of formulae. Laws of indices, exponentials and logarithms. (Formulae should be given to candidates so that the emphasis can be on their application). Linear, exponential and logarithmic equations. Changing the subject of formulae. 2. Types of data: qualitative, ordinal, quantitative discrete, quantitative continuous. Tabulation of data: contingency tables, frequency distributions, relative frequency distributions; cumulative frequency tables. Display of data: dot plots, stem and leaf diagrams, bar charts, line graphs, frequency and density histograms, frequency and density polygons, ogives and box plots. Symmetrical, skew and other distributions. Calculation: mean, mode and median from raw data and from frequency distributions; range, quartiles, inter-quartile range, variance and standard deviation; choice of statistics. 3. Need for sampling. Population, samples. Sampling frames. Sampling methods: census; simple random sampling; systematic sampling; multistage sampling; cluster sampling; stratified random sampling; quota sampling; convenience sampling. British Standards and their specification of appropriate sample sizes and sampling methods for specified quantities of various substances as a basis for agreement between supplier and purchaser. APPROACHES TO GENERATING EVIDENCE It is recommended that candidates are given real-life examples to illustrate the algebraic and statistical concepts and techniques used in this unit. These examples should be linked where possible to the quality management topics being studied. The design of the unit is such that it requires the use of a calculator with a statistical facility for direct calculation of mean and standard deviation. In practice only relatively small data sets can be dealt with using such a calculator or by hand drawn diagrams. If a statistical package is also available then much larger data sets can be studied, but these should be set up in advance. It is recommended that the consolidation of skills be achieved by including problem solving in a practical and vocational context and not only by mechanical exercises. Group investigations are to be encouraged. 7

8 Candidates are advised to maintain a workfile. This could comprise the candidate s own notes, handouts, worksheets, exercises, a logbook of computer activities and other relevant material. It is important that non-statistical terms are used to explain the significance of results. ASSESSMENT PROCEDURES Centres may use the instruments of assessment that are considered by tutors to be most appropriate. Examples of instruments of assessment that could be used are illustrated by exemplars. EXEMPLARS Outcome 1 Apply Algebraic Methods in problem solving PC (a) Round the following numbers to the specification given: (i) (ii) (iii) (iv) to 2 significant figures to the nearest hundred thousand to 3 decimal places to 4 decimal places PC (b) In sampling for detection, it is necessary to determine the sample size n necessary to have only a small chance (1 in b, where b is a whole number) of not detecting a rare but important defect which occurs in only 1 in x of the population, where x is also a whole number. The formula for n is: n 1nb = 1nx 1n ( x 1) Calculate n if b = 50 and x = 200 Q2 The probability that a component whose lifetime distribution follows a Weibull distribution will last for at least time t is F( t)=1 e t b a Where a is a shape parameter and b is a scale parameter. Calculate F(1000) for a distribution for which the shape parameter has value 0.65 and the scale parameter has value

9 PC (c) For components whose lifetime distribution follows an exponential distribution, the percentage P that will survive for at least t hours is P = 100e -λt where λ is a constant called the failure rate. For a component with a failure rate of hr -1, calculate the time t at which 95% still survive. Q2 In statistical process control, the capability index C p is defined by: C p = U L 6σ If U = mm and L = mm and the minimum acceptable value of C p is 1.33, calculate the value of σ that just achieves this value of C p. PC (d) Change the subject of the stratified random sampling formula n 1 = N1n N to n Q2 Change the subject of the formula s.e. = p( 1 p) n to n Outcome 2 Use Descriptive Statistics PC (a) - (i), Q2 (i) PC (b) - (ii), Q2 (ii) The percentage of washing machines from a production line that require readjustment on testing is recorded each hour for 24 hours with the following results:

10 (i) (ii) Form a stem and leaf diagram of the data using half stems. Order the leaves. Calculate the median and interquartile range of the data. Q2 At a milk bottling plant, a daily sample of 500 ml milk cartons is taken and the volume of milk in them recorded. On one day, 140 cartons were sampled with the following results. (Volumes are recorded to the nearest ml). Volume Contained (ml) Number of Cartons Volume Contained (ml) Number of Cartons (i) (ii) Draw a density histogram to illustrate the data. Calculate the mean and standard deviation of the data. Outcome 3 Select suitable methods of Data Collection PC (a) Select the most suitable method for choosing a sample for the following two situations from the list provided. Sampling methods: Simple random; systematic; stratified random; multistage; cluster; quota; convenience. Q2 A football association needs to plan its monthly random drug test of the players of the twelve clubs in its top league. The association requires to test a total of 30 players, and must take into account that there is a wide variation in the number of players contracted to each club. A survey into the heavy metal content of soils in Scotland requires data from a variety of locations. It requires data from each of the following classes of areas: industrial areas, urban areas, agricultural areas; with samples taken at distances: 2 metres, 10 metres and 50 metres from the side of a road in each area. 10

11 A further consideration is that the sample locations selection method should be chosen so that the amount of travel needed to collect the data is not excessive. PC (b) Your reports should concentrate on the statistical procedures and precautions necessary for collecting the samples; the appropriate sampling equipment should be stated but NOT described. If appropriate, the preparation of the laboratory sample from the material collected should also be described. Write a report on the procedures that would need to be carried out to sample a consignment of 120 cans, each containing 25 litres of liquid, in order to comply with: BS 4726 Sampling Raw Materials for Paints and Varnishes Q2 Write a report on the procedures that would need to be carried out to sample a consignment of a 16 tonne load, with maximum grain size 12 mm, and using the flattened - heap technique, in order to comply with: PC (c) BS 6043 Carbonaceous Materials Used in Aluminium Manufacture Part 4 Cold Ramming Pastes Section 4.1 Methods of Sampling A manufacturer has three plants producing tank loads of sulphuric acid. Plant A produces 63 tanks per day, Plant B produces 78 tanks, and plant C produces 122 tanks. To maintain a check on the purity of the acid, the manufacturer proposes to use a stratified random sample to take a combined total of 25 samples a day from the three plants. (i) (ii) (iii) Calculate the number of tanks to be sampled from each plant. Use random number tables to list the first two usable numbers for selecting tanks from Plant A to be sampled. Start at row 20, column 21 (ie, ). Use random number tables to list the first two usable numbers for selecting tanks from Plant C to be sampled. Start at row 31, column 3 (ie, ). 11

12 REFERENCES 1. Guide to unit writing, SQA, 1993 (Code: A018). 2. Guide to assessment, SQA, 1993 (Code: B005). 3. Guide to certification, SQA, 1996 (Code: F025). 4. Notes for unit writers, SQA, 1995 (Code: A041). For details of other SQA publications, please contact staff in the Sales and Despatch section (Tel: ) who can supply you with a copy of the publication list (Code: X037). Copyright SQA 1999 Please note that this publication may be reproduced in whole or in part for educational purposes provided that: (i) (ii) no profit is derived from the reproduction; if reproduced in part, the source is acknowledged. 12