WORKBOOK PROGRAMMING AND SUPERVISION OF CNC MACHINES

Transcription

1 WORKBOOK PROGRAMMING AND SUPERVISION OF CNC MACHINES LUBLIN 2014 Projekt współfinansowany ze ś rodków Unii Europejskiej w ramach Europejskiego Funduszu Społecznego

2 Author: Radosław Cechowicz Desktop publishing: Radosław Cechowicz Technical editor: Radosław Cechowicz Figures: Radosław Cechowicz Cover and graphic design: Radosław Cechowicz All rights reserved. No part of this publication may be scanned, photocopied, copied or distributed in any form, electronic, mechanical, photocopying, recording or otherwise, including the placing or distributing in digital form on the Internet or in local area networks, without the prior written permission of the copyright owner. Publikacja współfinansowana ze środków Unii Europejskiej w ramach Europejskiego Funduszu Społecznego w ramach projektu Inżynier z gwarancją jakości dostosowanie oferty Politechniki Lubelskiej do wymagań europejskiego rynku pracy Copyright by Radosław Cechowicz, Lublin University of Technology Lublin 2014 First edition Projekt współfinansowany ze ś rodków Unii Europejskiej w ramach Europejskiego Funduszu Społecznego

3 1 TABLE OF CONTENTS 1. THE NC PROGRAMMING LANGUAGE Selected functions of the NC programming language WORK STAND Description of the CNC control software Function codes handled by the software Programming the drilling cycles NC code format accepted by the machine Programming the milling cycles REPORTING THE RESULT OF THE PHYSICAL QUANTITY MEASUREMENT The rules for stating the result of the measurement The construction of the stem-and-leaf and the histogram diagram Deming's experiment DESCRIPTIVE STATISTICS AND THEIR PRESENTATION Construction of a histogram. Box-plot diagrams Pareto analysis PROPERTIES OF A NORMAL DISTRIBUTION AND ITS USAGE The Central limit theorem Estimating of the non-conforming fraction. Process capability TEST OF THE STATISTICAL HYPOTHESES Determination of the sample size; Power of a statistical test;...17

4 2 1. THE NC PROGRAMMING LANGUAGE 1.1. Selected functions of the NC programming language NOTE: Bold font has been used for the commands defined in ISO6983-1:2009 Default power-on commands according to ISO6983-1:2009 are on grey background. Turning Milling Description Example M 1 Program and block numbering O O Program number O0122 N N Block number N Machine measurement units, coordinate system and reference points (program zero) G53 G53 Program reference point reset G53 M1 G54, G55, G54, G55, G56...G59 G56...G59 G54 M1 G50, G92 G92 Sets program zero coordinates; used in machines that do not have G54..G59. Works like G54...G59 and G41..G44 G92 X10 Y20 Z15 L G20, G70 G20, G70 Inch mode (coordinate values are in inch) G20 M2 G21, G71 G21, G71 Metric mode (coordinate values are in milimeters) G21 M2 G90 G90 Absolute mode (all dimensions refer to program zero) G90 M3 G91 G91 Incremental mode (dimension chains) G91 M3 -- G17 Sets work plane to XY G17 M4 -- G18 Sets work plane to XZ G18 M4 -- G19 Sets work plane to YZ G19 M4 Cutting parameters G96 -- Constant cutting speed (Cutting speed value in m/min or ipm is set using the S command) G96 S150 M5 G97 -- Constant spindle rotating speed (Rotating speed value in rpm is set using the S command) G96 S2000 M5 G93 G93 Programs feed speed by setting the time of the operation (Inverse time). Ex: G93 F1 sets operation time to 1min G93 F10 M6 G98 G94 G94 Feed set in mm/min or ipm G98 F100 M6 G99 G95 -- Feed set in mm/rotation or ipr G99 F100 M6 1 Function type according to ISO6983-1:2009: M Modal after calling remains active until modified or replaced by another command from the same group (groups are designated with numbers M1, M2, etc.) L not modal active only in a block where called

5 3 Turning Milling Description Example M F -- Feed value (used in conjunction with G98/G99) G98 F100 M -- F Feed value (in milling centres in mm/min or ipm) F100 M S -- Spindle speed value (used in conjunction with G96/G97) S1300 M -- S Spindle speed value (in milling centres usually in rpm) S1300 M T -- Tool change (followed by tool nr and wear offset nr) T0202 M -- T Tool change (followed by tool number) T2 or T2 M06 Tool movement - basic G00 G00 Rapid motion G00 X15.2 Z1.5 M7 G01 G01 Linear motion (linear interpolation; used with F) G01 X1.5 Y12.0 F80 M7 G G02 G G03 Circular interpolation, clockwise (used with F and appropriate parameters, ex: X, Z, R or X, Z, I, K) Circular interpolation, clockwise (used with F and appropriate parameters, ex: X, Y, R or X, Y, I, J) Circular interpolation, anticlockwise (used with F and appropriate parameters, ex: X, Z, R or X, Z, I, K) Circular interpolation, anticlockwise (used with F and appropriate parameters, ex: X, Y, R or X, Y, I, J) G02 X2.0 Z4.0 I-12.5 K2.5 or G02 X5.0 Z4.0 R5.0 G02 X2.0 Y4.0 I-12.5 J2.5 or G02 X5.0 Y4.0 R5.0 G03 X2.0 Z4.0 I-12.5 K2.5 or G03 X5.0 Z4.0 R5.0 G03 X2.0 Y4.0 I-12.5 J2.5 or G03 X5.0 Y4.0 R5.0 G04 G04 Dwell. Dwell time can be defined by parameter X, F or P G04 X2.0 (2s) (units: seconds or milliseconds) or G04 F20 (20ms) M7 G06 G06 Parabolic interpolation. Parameters I, J, K are used to G06 X2.0 Y4.0 I-12.5 define vertex coordinates J2.5 M7 G33 G33 Threading, constant pitch. M7 G34 G34 Threading, increasing pitch. M7 G35 G35 Threading, decreasing pitch. M7 Tool movement - advanced G09 G09 Exact stop. Operation ends after the tool comes to a complete stop. Used to improve accuracy, slows down G09 L the program execution. NON-MODAL version. G60 G60 Exact stop function. Operation ends after the tool comes to a complete stop. Used to improve accuracy, slows G60 M8 down the program execution. MODAL version. G64 G64 Exact stop reset (cancellation) see G60 G64 M8 G63 G63 Sets the threading mode. In threading mode the Feedrate Override control on the operator's panel is disabled, feed cannot be controlled manually. Used for threading with taps. G63 L M M7 M7 M7 M7

6 4 Turning Milling Description Example M Resets tool radius compensation (see G41, G42) According to ISO6983-1:2009 this function also resets G40 G40 tool length compensation (see. G43, G44). G40 M9 Fanuc controllers use G49 to reset toll length compensation. G41 G41 Sets tool radius compensation (cutting on the right side of tool path). Tool radius is read from the pre-set D G41 D... M9 register. G42 G42 Sets tool radius compensation (cutting on the left side of tool path). Tool radius is read from the pre-set D register. G42 D... M9 G43 G43 Sets tool length compensation (toll length is a positive number). Tool length is read from the pre-set H register. G43 H... (M9) G44 G44 Sets tool length compensation (toll length is a negative number). Tool length is read from the pre-set H register. G43 H... (M9) G49 G49 Resets tool radius compensation in Fanuc controllers (see G40) G49 (M9) Fixed cycles 2 Machine reference point return (tool change point G28, G74 G28, G74 return) G91 G28 Z0 L G80 G80 Cancels all fixed cycles (see G81..G89) G80 M10 G81 G81 Sets the fixed cycle drilling M10 G82 G82 Sets the fixed cycle drilling with dwell (counter-boring) M10 G83 G83 Sets the fixed cycle deep hole drilling (with tool withdrawal) M10 G84 G84 Sets the fixed cycle tapping M10 G85 G85 Sets the fixed cycle boring (rough) M10 G86 G86 Sets the fixed cycle boring with dwell (rough) M10 G87 G87 Sets the fixed cycle boring (finishing) M10 G88 G88 Sets the fixed cycle boring with dwell (finishing) M10 G89 G89 Sets the fixed cycle reaming M10 Machine functions 3 M00 M00 Stop (unconditional) M00 A L M01 M01 Optional Stop (active only if Optional Stop switch in ON position) M01 A L M02 M02 Program end. Stops the spindle and other devices (like coolant pump). Used for machine reset. M02 A L 2 Application examples are presented during classes 3 Function type according to ISO6983-1:2009: A Function activated after tool stops (tool movement completed before function) B Function activated parallel to tool movement (function engaged during tool motion) M Modal active until modified or cancelled by a function from the same group L Not modal active only in the block where was called

7 5 Turning Milling Description Example M M03 M03 Spindle start clockwise M03 M M04 M04 Spindle start anticlockwise M04 M M05 M05 Spindle stop M05 M M06 M06 Tool change. Coolant pump may be switched off on some machines. M06 L M07 M07 Coolant on (mist) M08 M08 Coolant on (coolant pump on) M08 M09 M09 Coolant off M09 M10 M10 Material hold (engage material holding system) M10 L M11 M11 Material release (disengage material holding system) M11 L M20 -- Tailstosk disengage M20 M21 -- Tailstock engage M21 M30 M30 End of data. Like M02 but machine returns to the beginning of the active program (so it can be re-started with green button) M30 M41 -- Spindle speed ranges sets the first speed range (usually used for rough machining) M41 M42 -- Spindle speed ranges sets the second speed range M42 M43 -- Spindle speed ranges sets the third speed range M43 M44 -- Spindle speed ranges sets the fourth speed range M44 M48 M48 Enables spindle speed and feedrate control with the operator's panel controls (see M49) M48 M49 M49 Disables spindle speed and feedrate control with the operator's panel controls. M49 A B M60 M60 Pallet change or part setup change. Stops spindle and coolant M60 L M98 M98 Sub-program call (U program number, L number of program M98 U123 L3 executions) M99, M17 M99,M17 Sub-program end return to the main program M99 Notes: A L

8 6 2. WORK STAND General view of the work stand with the C-type milling machine is shown on Fig. 1. spindle controller C-frame milling machine main switch control computer emergency switch axis controller Fig. 1:Work stand with the CNC milling machine 2.1. Description of the CNC control software The functions of the cnc software, which will be used during the laboratory exercises can be accessed from the main screen (Fig. 2), the programming screen (Fig. 4) and the manual operations screen. The function keys in the main screen have the following assignments: Esc (koniec) end of task (exits the program or returns to main screen) F1 (pomoc) help (in Polish) F2 (programy) access to file operations menu (also allows to create new program) F3 (parametry) edit or manage machine parameters (like the scaling factor) F4 (inne funkcje) )- other functions F5 (bazuj) homing (must be executed after the machine is switched on) F6 (wykonaj) execute the active program (program must be loaded with F10 first)

9 7 F7 (reczna) manual control mode (see Fig. 3) F8 (symuluj) program simulation mode (program must be loaded with F10 first) F9 (popraw) edit the active program (quick alternative to F2) F10 (ładuj) choose and load the program into machine memory current tool position active program name, scaling factor coordinates of the program zero machine coniguration parameters function keys Fig. 2: Main screen of the controlling program (before homing the machine) current tool coordinates axis speed value axis speed controls Y axis controls (Left/ Right arrow) Z axis control (Page Up/Page Down) Tool change menu X axis control (Up/ Down arrow) Fig. 3: Manual control screen Spindle control (I nsert) program edit operations create new program Fig. 4: New program window (available after pressing F2 on the main screen). Program editing window becomes available after the creation of a new program or after pressing F9 on the main screen.

10 8 machine zero point simulation screen menu machine workspace (green rectangle) axis Z range (of the active program) position of program zero on Z axis axes X and Y range (of the active prgm) program zero position on XY plane Fig. 5: Main screen of the simulation mode (can be accessed from the main screen after pressing F8). After the NC code has been written, the program has to be tested with the simulation functions (main screen: Fig. 5). The following functions are available in the simulation screen menu: PodPliku - the preview of all the tool paths (the entire program) DefMater - sets of the size and position of the material (a grey rectangle on the preview screen) DoMater - sets the program zero point in the upper-left corner of the defined material (all axes) DefOffset - manual entering of the coordinates of the program zero point ZerOffset - sets the program zero point in the machine reference point (machine zero; all the axes) Stan - screen displaying the information on the range of the axes movement (Fig. 7) TrajPow - draws the programmed tool path (full screen) TrajRze - draws the programmed tool path (against the machine work area) Symuluj - simulation of the machining (requires the correct setting of the work-piece dimensions and program zero point) Kolory - configuration of colours assigned to particular tools Koniec quits the simulation mode Function codes handled by the software Machine functions Preparatory functions M0, M1 - program stop, M2, M30 - program end, M3, M4 - spindle on M5 - spindle off M99 - end of the sub-program F - feed rate in mm/min T - tool change % - this symbol is obligatory at the beginning and at the end of each program G0 rapid motion G1 linear interpolation G2, G3 circular interpolation G4 dwell (argument: code X [ms]) G22 sub-program call (P ) G40 tool radius compensation off G41, G42 tool radius compensation G80 cancel canned (fixed) cycle G90, G91 absolute and incremental positioning Fixed cycles: G61, G77, G78, G79, G81, G82, G83, G87, G88, G89 Full description of the CNC software can be found in the documentation of the machine.

11 Programming the drilling cycles The milling machines in the CNC machine laboratory have the following fixed cycles: No. G-code Name Description 1 G81 Drilling Syntax: G81 Z... W... Z depth of a hole relative (incremental) negative value, calculated from the retraction plane (see Fig. 6). W distance to the retraction plane positive incremental value measured from the initial tool position (see Fig. 6). 2 G82 Chip-breaking drilling 3 G83 Peck drilling (with breaking and removal of chips) Syntax: G82 Z... W... B... D... K... B time for chip breaking in seconds D value by which successive K-steps are decreased (positive) K length of a single drilling step between the successive intervals for the chip-breaking (positive; see Fig. 7) Other parameters as in G81 Syntax: G83 Z... W... B... D... K... A... A time for chip removal in seconds (tool dwells at the retract plane) Other parameters as in G81 and G82 initial tool position initial plane Z B Z B Z R W rapid motion traverse tool retraction plane Z R Z machining with programmed feed Z S Fig. 6: Initial and tool retract planes in fixes cycles programming programmed end position Z S Narzędzie initial w pozycji tool position początkowej Płaszczyzna initial początkowa plane Z Z B B Z B Przejazd rapid ruchem motion przestawczym traverse W Z R Z S Z K K-D K-D Płaszczyzna retract plane wycofania Z Z R R Pierwszy drilling/ milling przejazd depth ruchem (irst roboczym tool entry) (K) Kolejny drilling/ przejazd milling depth ruchem (successive roboczym entr.) (K-D) programmed Pozycja końcowa end position Z S Z S Fig. 7: Programming the chip-breaking cycle and pocket machining

12 10 No. G-code Name Description 4 G79 Execution of a programmed cycle in a single point 5 G78 Execution of a series of programmed cycles along a straight line 6 G77 Execution of a series of programmed cycles along an arc Syntax: G79 X... Y... X, Y coordinates of the point in which the hole is to be made Syntax: G78 X... Y... I... J... S... or G78 X... Y... A... D... S... X, Y coordinates of the first hole to be made I, J relative distance to the next hole along the X and Y axes A - inclination angle of a line in relation to axis X (positive in the 1st and the 2nd quadrant) D distance to the next point along the line S number of repetitions (number of holes to be made) Syntax: G77 X... Y... A... B... D... S... X, Y coordinates of the centre of the arc (relative to the program zero point) A angle between axis X and the radius indicating the first hole B the radius of the arc on which the holes are to be made D angular distance (in degrees) between the succeeding holes S number of repetitions (number of holes to be made) 2.4. NC code format accepted by the machine When writing the program in the NC language, the following rules have to be respected: A program has to begin and end with the "%" symbol. At the end of the program, the line before the "%" symbol has to contain the "M30" command, which ends the program; previously, the spindle has to be turned off with the "M5" command. At the beginning of the program, a tool has to be set (even if it has already been mounted in the spindle; the "T " command) and moved along all the axes (X, Y and Z). It is recommended to move the tool to the program zero point on the X and Y axes and up to the safe position over the material on the Z axis. Whenever radius compensation is used, the first and last sections of the tool path should be programmed as straight lines (G0 and G1 commands). Arcs can be programmed through setting the radius "R" (positive or negative value) or the coordinates of the centre (I and J; the coordinates should be absolute).

13 Programming the milling cycles Y I G87 G88 G89 X Fig. 8: Pocket milling cycles available on the machines in the laboratory No. G-code Name Description 1 G87 Rectangular pocket milling 2 G88 Circular pocket milling 3 G89 Milling of a circular pocket with a bos Syntax: G87 X... Y... Z... I... K... W... X, Y dimensions of a pocket along the X and Y axes Z depth of a hole relative (incremental) negative value, measured from the retraction plane (as in Fig. 6). W distance to the retraction plane positive value measured incrementally from the initial tool position (as in Fig. 6). I depth of cut on the XY plane in the percentage of the width of the tool positive value (as in Fig. 8). K depth of cut along the Z axis positive incremental value as shown on Fig. 7 Syntax: G88 B... Z... I... K... W... B radius of the pocket Other parameters as in G87 Syntax: G89 B... C... Z... I... K... W... C radius of the bos Other parameters as in G87 and G88 Notes

14 12 3. REPORTING THE RESULT OF THE PHYSICAL QUANTITY MEASUREMENT 3.1. The rules for stating the result of the measurement Problem 1: The result of the measurement indicates that the frequency of string vibrations equals 210 Hz. The uncertainty interval for the measurement was established as the following: Hz. Write down the result of the measurement using the standard form. Problem 2: The result of the measurement determined the vibration frequency to be 230±15Hz. Establish the uncertainty interval for the given outcome. How to interpret the record of the measurement result? Problem 3: Write down the uncertainties of the following measurements: the best approximation uncertainty interval 34,8 km od 34,2 do 35,4 km 23,12 MPa od 20 do 26 MPa 112 V od 108 do 116 V Problem 4: Write down the following results of the measurements in a proper form: p = 7, ± 0,02345 MPa; x = 4,2345*10 4 ± 2 m; q = 2,45678*10-7 ±3*10-9 F; x = 0, ± 0, m; p = 4,345* 10 3 ±22 kpa; t = 1,6234 ± 1 s; t = 3,8932 ± 3 s; 3.2. The construction of the stem-and-leaf and the histogram diagram Problem 1: The results of the measurements are organised into a table (Table 1) and show the percentage of cotton in a textile material. Design the stem-and-leaf display and, on the basis of the obtained diagram, determine the mean value, median, upper and lower quartiles and the interquartile range. Verify the results of the calculations using the STATISTICA software.

15 13 34,2 37,8 33,6 32,6 33,8 35,8 34,7 34,6 33,1 36,6 34,7 33,1 34,2 37,6 33,6 33,6 34,5 35, ,6 33,4 37,3 32,5 34,1 35,6 34,6 35,4 35,9 34,7 34,6 34,1 34,7 36,3 33,8 36,2 34,7 34,6 35,5 35,1 35,7 35,1 37,1 36,8 33,6 35,2 32,8 36,8 36,8 34, ,1 32, ,1 37,9 34,3 33,6 34,1 35,3 33,5 34,9 34,5 36,4 32,7 Table 1: Measurement results of the percentage of cotton in a textile material Deming's experiment Problem 1: Create a new variable of a Statistica spreadsheet and name it as D. Using the RndNormal(x) function of the Statistica software, generate a sequence of 100 pseudo-random numbers, whose mean value is 50 and standard deviation equals 2. Afterwards, generate a sequence of numbers that constitutes the difference between the mean value (50) and the previously generated variable. Record the result of the calculations in the next variable DR symbol. Calculate the backward differences for the original D sequence and DR variable, and assign the result to the next variable W, then calculate its standard deviation. Which of the considered variables D and W shows greater dispersion? Explain the obtained result. 4. DESCRIPTIVE STATISTICS AND THEIR PRESENTATION 4.1. Construction of a histogram. Box-plot diagrams Problem 1: Multiple measurement samples of the post-mould shrinkage of the moulded pieces are organised in a table (Table 2). Complete the following steps: Using the STATISTICA software, prepare a box-plot diagram, which compares the results of the measurements obtained in particular samples s1 s10. Then, make a consolidated histogram including all the results of the measurements (s variable). Compare the box-plot diagrams for s1 s10 variables with the result of the histogram for the s variable. Is there a substantial difference between the results of the measurements? Calculate the mean value and standard deviation for particular s1 s10 variables. Afterwards, calculate the mean value of means and the standard error. Compare the obtained results with the result of the calculation of the mean value of the s variable. Comment on the obtained results.

16 14 Assess the goodness of fit of the distribution of the s variable measurements to the normal distribution using the probability-probability plot; Establish the fraction of defectively manufactured moulded pieces, assuming that the distribution of the post-mould shrinkage fits the normal distribution. Assume that the lower and upper specification limit equals 3.2 and 7.7 (%), respectively. S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 4,5 6,2 5,4 4,8 3,6 4,7 5,1 4,8 4,4 5,5 5,1 4,2 5,0 5,4 7,1 6,1 6,0 5,5 6,4 6,5 4,3 4,2 4,1 5,0 3,7 5,8 5,7 5,7 6,1 4,7 4,6 4,3 6,5 7,7 4,4 4,9 4,9 4,6 6,1 5,6 5,7 4,2 6,6 4,8 5,0 6,8 3,7 3,9 4,9 5,9 5,5 4,6 4,7 6,0 6,4 7,0 6,1 6,5 5,1 5,5 6,1 3,8 5,7 6,7 6,3 5,2 5,5 6,5 5,2 7,5 5,1 4,0 7,0 5,8 7,0 5,9 6,0 5,6 5,5 4,2 6,5 7,3 5,9 6,0 7,3 5,4 4,6 6,6 5,4 3,8 5,3 6,3 5,7 5,4 5,6 5,8 4,7 4,9 6,3 3,6 Table 2: Measurement results of the post-mould shrinkage of the moulded pieces. Comment: a table header row shows a symbol of a sample, while a corresponding table column organises the results of this sample Problem 2: Multiple measurement samples of the voltage on the battery terminals are organized into a Table Table 3. Prepare a histogram (empirical distribution) of the analysed variable and then calculate and mark the following descriptive statistics: the mean value, median, mode and the upper and lower quartile. A 1 A 2 A 3 A 4 A 5 A 6 12,62 12,74 12,7 12,63 12,63 12,66 12,59 12,68 12,79 12,68 12,65 12,56 12,59 12,63 12,61 12,56 12,65 12,66 12,68 12,61 12,75 12,56 12,62 12,71 12,66 12,59 12,68 12,59 12,71 12,73 12,68 12,67 12,63 12,57 12,64 12,6 12,64 12,62 12,73 12,64 12,67 12,71 12,67 12,69 12,58 12,62 12,61 12,73 12,64 12,65 12,61 12,69 12,63 12,61 12,66 12,67 12,63 12,61 12,69 12,72 Table 3: Measurement results of the voltage taken on the battery terminals. Comment: a table header shows a symbol of a sample, while a corresponding table column organizes the results of this sample

17 Pareto analysis Problem 1: The results of the moulding process observation were presented in relation to the registered defects and the expenses borne during the production of the moulded pieces (see Table Table 4). Fill in the table columns (percentage of defects, percentage of expenses). Given so, prepare two independent statements: the first concerning the percentage of defects and the second regarding the percentage of the expenses, according to the defect type. In order to reduce the loss sustained during the manufacturing, take advantage of the Pareto Analysis and establish which of the defects should be eliminated first? Type of defect Flaw description No. of defects [-] Perc. of defects [%] Expenses of materials, [zł] Expenses of labour [zł] Expenses of manufacturing [zł] Total expenses [zł] 1 underflowing 59 11,8 7,25 23,52 42,57 Perc. of expenses, [%] 2 tension ,67 15,14 48,69 258,5 3 line of flows ,8 11,08 35,43 228,31 4 shrinkage 2 0,38 1,14 3,64 5,16 5 overflowing 60 11,4 8,08 26,34 45,82 Table 4: Results of the moulding process observation 5. PROPERTIES OF A NORMAL DISTRIBUTION AND ITS USAGE 5.1. The Central limit theorem Problem 1: Launch the program named "The Central Theorem Limit.exe", attached to the script, familiarise yourself with the content of the exercise and carry out the following instructions: 1. Choose the normal distribution form the "Population distribution" list. Select 5 as the number of measurements in a sample. Next, choose the mean value as the statistic and generate 1000 samples (measurements). Are the mean value of means and the standard error consistent with the population parameters? What can be thought of the obtained distribution of the sample statistics. 2. Repeat the instruction (1) but choose the standard deviation (SD) as the analysed statistic and generate 1000 samples. Is the mean value of the calculated SD statistic consistent with the value of the population parameter?

18 16 3. Repeat the instruction (1), though change the distribution function to the exponential distribution. The number of measurements in a sample is to be n = 5. What is the distribution form of the values of the statistic from the sample?; 4. Repeat the instruction (3), though increase the number of measurements in a sample to 20; 5. Repeat the instruction (3), though increase the number of measurements in a sample to 20 and, simultaneously, change the form of distribution to the uniform one; 6. Repeat the instruction (5), though increase the number of measurement in a sample to 100. Interpret the obtained results of the calculations Estimating of the non-conforming fraction. Process capability Problem 1: In many cases of the industrial use of the SPC, it is required to estimate the proportion of the production which does not meet the expected specification. Assume that the technological process consists in dividing the product and final packaging of its portions. 100 portions of this product were randomly chosen and weighed. It turned out that the observations belonged to the normal distribution (Fig. 9). The mean value of the measurement was 255g, while the standard deviation equalled 4.73g. 250±10 g was adopted as the upper and lower specification of a single portion. How many portions were produced outside the limits of specification? How will the non-conforming fraction change if the mean value of the portions is consistent with the aim of the process? y = n o r m a l( x ; ; 4, 7 3 ) 0, 0 8 0, 0 6 0, 0 4 0, 0 2 0, Fig. 9: Probability distributions illustrating the measurement results with the limits of specification being highlighted

19 17 6. TEST OF THE STATISTICAL HYPOTHESES 6.1. Determination of the sample size; Power of a statistical test; Problem 1: The melting point measurement carried out on n=10 units of the connectors used in the fuel production process equalled T=154.2 o C. It was established that the temperature measurements belong to the normal distribution with the standard deviation σ=1.5 o C. 1. Carry out a statistical test comparing the results of the measurement of the connectors melting point with the referential value T 0 =155 o C, assuming the value of I-type error, α=0.05, as defining for the level of statistical significance. How to interpret the test results? 2. What is the value of the probability P of the conducted test; 3. What is the value of a second-type error β, assuming that the population mean is μ = 150?; 4. What would be the required n sample size if the II-type error was limited by a condition: β<0.1. Assume that the I-type error equals α=0.05. Problem 2: Assume that the quality characteristic R describes the wear of the new construction of the drive-train element, measured after km mileage, and is an important aspect associated with the warranty repairs. 17 drive-train units were examined and the mean value was determined to be 4.42 (units). The previously performed tests proved that the analysed quality characteristic belongs to the normal distribution and the standard deviation value equals σ=0.7. On the basis of the given information, carry out a statistical test in order to find out whether the drive-train new construction meets the requirements included in the design (specification) of the drive-train R 0 =3.8. Assume that the I-type error equals α=0.05. What is the power of the test? What should be the sample size if a quality engineer would like to detect a change in a construction wear dr = 1.25 σ (units). The power of the test is assumed to be 0.95?