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Ventilation Lab #2 Airflow Quantity Measurements January 27 & 29, 1999 Purpose The purpose of this lab is to familiarize the students with the UMR Experimental Mine ventilation facilities, to select stations for ventilation readings, to measure the cross-sectional area of the selected stations, and to measure the distance between the selected stations. The second portion of the lab is to familiarize the students with the common velocity measuring instruments, to utilize these instruments in making an airflow survey and to determine airflow directions at measuring stations throughout the Experimental Mine. This lab is an introduction to the Experimental Mine's ventilation system and to gather preliminary data for future labs. Procedure 1) 2) 3) 4) 5) 6) 7) Meet at mine classroom to discuss the lab. Review the instruments to be used. Tour the mine, noting the location of stoppings and shafts. Select the stations. Measure station cross-sections and separation distances. Take velocity measurements at the ventilation stations. Determine airflow through room-and-pillar area of the mine. Laboratory Instructions Locate ventilation stations away from intersections, areas of non-uniform cross-section and any sudden changes in cross-section. These items tend to cause turbulent airflow which will reduce the accuracy of readings. It is also important that permanent ventilation stations be used. This will save time and give continuity to the readings for the later laboratories. The cross-sectional area of the drift will be calculated by measuring the height and the width at three points and averaging the values. The points selected for measurement should be representative of the drift at that location. Measuring stations will be marked with a 'V' on the wall of the drift with consecutive numbers (for example, station 1 is V-1, station 2 is V-2 and so on). The distance between stations will be measured from V-1 to V-2 to V-3 etc. The measurements will be made along the centerline of the drift. Min-218 Lab #2 Air Quantity Measurement Vane anemometers, smoke tubes and a Kurtz hot wire anemometer will be used to make velocity measurements at the previously designated ventilation stations. The majority of the measurements will be made with the vane anemometer utilizing a traverse method. At appropriate locations, the hot wire anemometer and smoke tube will be used. Instrument Used Vane Anemometer: all stations Hot Wire Anemometer: (demonstration only) Smoke Tube: (demonstration only) Report The report will contain a map of the mine with stations, regulators and shafts clearly marked. A table of results should be similar to: Station # 1 1-2 2 ... Sep. Distance Ave. Height 0.00 Ave. Width 0.00 Area 0.00 0.00 ... 0.00 ... 0.00 ... 0.00 ... The report will also consist of calculations of air velocities and quantities at all measuring stations. A comparison should be made between the various measuring instruments where applicable. Leakage through stoppings should also be estimated from the data where applicable. The map should also include details of the airflow direction in the mine. Differences between areas of high velocity and low velocity should be clearly marked. Furthermore, a clear indication of the direction of leakage through all stoppings should be included. All other data and results should be tabulated in a neat and concise format. Information Using the data from this lab the quantity of air movement at each station can be found. Quantity = Velocity x Area Q = VA Stoppings (or anything used to direct air movement) can be made of various materials. In the Experimental Mine, plastic, brattice cloth, and Kennedy stoppings are used. In other mines concrete blocks, wood, and/or metals have been used depending on the situation: cost, amount of activity in the area, ventilation requirements, and others. Air velocities are classified into three ranges: (1) low air velocity [less than 100 fpm], (2) medium air velocity [between 100-750 fpm], and high air velocity [greater than 750 fpm]. Three instruments are used in this lab to cover all three of these ranges. 2 Min-218 Lab #2 Air Quantity Measurement 1. Vane Anemometer (medium-high) is one of the most commonly used instruments. It has good accuracy when used correctly. 2. Hot-wire Anemometer (low-high) is highly accurate and easy to use, but is more expensive and requires more care. 3. Smoke Tube (low) is simple and easy to use but it is not very accurate. Mostly used to tell general information about the air movement in an area. However, at rates of less than 5025 fpm it becomes a more important tool in measuring velocity. These instruments are used in one of three ways: 1. Point Method is where the opening is divided into equal sections (such as 9 or 16), and an instrument is placed in the center of each section for a measured time period or until the instrument stabilizes. Then the readings are read directly (ft/min. or ft/s) and averaged or the distance recorded (ft) on the instrument is divided by the time (sec) and then averaged. The type of instrument used determines the calculations needed. 2. Continuous Traverse Method is most often associated with a vane anemometer. In this method the instrument is moved across the entire opening in an up and down continuous traverse. This traverse is timed, and by dividing the distance recorded (ft) by the time (s) the velocity is found. 3. Smoke Tube Method in this method a puff of smoke is released at one point and is timed until it reaches another point. The smoke should be released where the velocity is the highest in the opening (this is usually the center). With a time (s) and distance (ft) a velocity can be figured. This velocity is for the highest rate of air movement in the opening. To compensate for varying velocities across the opening the velocity found is multiplied by 0 . 8 . This new velocity is an approximate velocity and for more precise measurements another method should be used, but for rough ideas of air movement it is simple and quick. Notes 1. Give lengths in feet (5.55') not in inches when conducting calculations. 2. Use best judgment in measuring the opening. Do not measure the opening right next to the back (roof) or floor. Break the opening into equal parts. Note which reading is which, the right, center, left, top or bottom. 3. For table of results, use the example provided but sum the separation distances as you proceed. Example Station # 1 2 3 4 Distance –– 21' 44' 29' Cumulative Distance –– 21' 65' 94' 4. When using a vane anemometer in a continuous traverse, use the extension rod. Make sure everyone else is down wind. The traverse should consist of about three downward passes and 3 Min-218 Lab #2 Air Quantity Measurement two upward passes for one minute. Speed of traversing depends on the velocity measured and specific situations at measuring site. At least two passes should be made starting on opposite sides of the opening. 5. With the point method, again make sure everyone is down wind. Divide the opening into nine areas to take readings. 6. When using a smoke tube do not let anyone move for a short time before taking the reading, to help cut down on turbulence. Release the puff of smoke perpendicular to the drift to prevent any added velocity to the smoke. The time is counted from when the smoke is released until the main body of smoke, reaches the second point. ADDITIONAL INFORMATION ON ACCURATE AIRFLOW MEASUREMENTS 1 . General To ensure an effective, efficient, and economical mine ventilation system, reliable data must be obtained on the various properties of mine air. Such data then form the basis for analysis, evaluation, planning and subsequent recommendations for improvements or changes that may be required. The most important properties of mine ventilation are air velocity and ventilation pressure. When these properties are known at various strategic points in the mine, the overall quantity control problem can be resolved by utilizing analytical and/or numerical methods of fluid. The accuracy of the air quantity measurement underground depends on many factors. The three most important factors are accurate velocity and cross-sectional area measurements. Air quantities are determined by cross-sectional area the air is passing through and the air velocity. Accurate and representative areas can be obtained through diligent and careful practices using measuring tapes. Air velocities are determined using anemometers, which are then used to determine air quantity. Altimeters and barometers are used for pressure measurements. 2 . Air Quantity Measurements Airflow quantity is calculated through the product of the average air velocity and the cross-sectional area of the air passage. Typically, the mine ventilation engineer has obtained the average air velocity with a time integrating vane anemometer and a stopwatch. These are employed over a continuous traverse method which includes a continuous sweep either horizontally or vertically (Figure 1). Air velocity is measured using a vane anemometer. It utilizes the kinetic energy of an airstream to drive its windmill like impeller. The rotation of the impeller blade is proportional to the air velocity and translation of the impeller rotation speed will give a measure of the air velocity. They are rugged, affordable, easy to use, and give fairly accurate and reliable results if precautions are taken while making measurements.1 During the last two decades there have been various stages of de1 With suitable precautions, a vane anemometer can give repeat measurements with a precision of ± 2.5 percent according to several studies. 4 Min-218 Lab #2 Air Quantity Measurement velopment each introducing the latest technology of the time up to the present day with the inclusion of micro-electronics and processors. Historically, this has been facilitated by a mechanical analog instrument that has either a counter or set of dials that display the flow. Common instruments of this type are Davis', or Taylor's Biram type anemometer, or Airflow Development's model AM5000. Both Taylor and Davis are widely used in U.S. mining industry but unfortunately, Taylor is no longer available. Just like its counterpart in U.S., AM5000, which is widely used in the Canadian mining industry, is also no longer available. End Start Start End Figure 1 Methods of measuring velocity in mine airways Over the last two decades there have been numerous advances in the development of electronic air velocity meters as well. Currently available units have the following features: (1) battery operation; (2) are compact and can be hand-held; (3) employ LCD's, and microelectronic circuitry; and (4) provide a time integration for the duration of a traverse. These features make such instruments a possible replacement to the analog units and stopwatch determinations. 3 . Making Accurate Air Velocity Measurement Depending on the velocities measured, different types of anemometers are used. For measuring velocities from 100 to 2,000 fpm, ordinary medium-velocity vane anemometers are usually the most commonly used. To determine air velocities below 100 fpm, either a low-speed anemometer or smoke tube can be used. For high air velocities, special high-speed anemometers (usually equipped with half as many vanes as the standard type to avoid bearing cracking) or pitot tubes can be used. We will be emphasizing the anemometers since most air velocities underground fall between 100 to 2,000 fpm range. Please refer to Attachment II for further information on anemometer application. Low Velocity Measurement Either low velocity anemometers or smoke tubes can be used for measuring low velocities. For smoke method, the three items of equipment needed are a smoke tube, a watch, and a tape. The sequence is as follows: ❐ Pick the best smooth-walled, straight, and unobstructed section of airway available away from bends upstream. ❐ Measure off and mark a distance for the smoke cloud to travel. 5 Min-218 Lab #2 Air Quantity Measurement The distance varies depending on how well the smoke cloud hangs together. 20' is usually adequate for fairly rapidly moving air while 10' might be maximum for very slow moving air. One person holds the watch at the downstream end of the measured distance and the second person holds the smoke tube at the upstream end. At the moment the smoke is released, the timing is started. Travel time is measured when the leading edge of the smoke cloud passes the downstream mark. Smoke visibility is improved if the upstream person, only, shines a light on the smoke cloud. Smoke cloud release points are generally spotted top, middle, and bottom for each of the left, center, and right sides of the drift. More readings are made if better accuracy is desired. The area of the drift is then measured, usually with three vertical measurements and three horizontal measurements (right, center, and left; and top, middle, and bottom). Again, for better accuracy, more measurements may be taken. The intent is to get the best figure for the velocity of the smoke cloud from an average and the best figure for the area that the smoke cloud travels in. Of the two types of measurements made, it has been found that the area is the more difficult to determine accurately. Example 1: The smoke tube method is used to determine the air quantities in an airway. The data is as follows: Air traveling distance: 20' Travel time in minutes: 0.15, 0.14, 0.14, 0.14, 0.13, 0.16, 0.14, 0.15, 0.13. Beginning area: height - 7.1', 8.1', 7.3' width - 6.1', 7.0', 7.9' Ending area: height - 7.5', 8.6', 8.1' width - 6.1, 7.0', 7.9' Solution : 0.15+0.14+0.14+0.14+0.13+0.16+0.14+0.15+0.13 Average travel = 9 = 0.142 minute Velocity = 20' ÷ 0.142 = 141 fpm Beginning area = ave. height x ave. width = 7.5' x 7.07' = 53.02 ft2 Ending area = ave. height x ave. width = 8.07' x 7.00' = 56.49 ft2 Average area = (56.49 + 53.02)/2 = 54.76 ft 2 Q = AV = 54.76 x 141 = 7,721 cfm ≈ 7,720 cfm Medium Velocity Measurement Anemometers are usually used for medium velocity measurement. Alternate equipment may be a swinging vane velometer, a vortex anemometer, or (rarely) a hot wire or hot film anemometer. The procedure is as follows: ❐ Select the best smooth-walled, straight, and unobstructed section of airway available away from upstream. ❐ With the anemometer attached to an extension bar (or a wand), make a one-minute zig zag slow moving traverse across the airway from top to bottom, moving clear across from one side to another. ❐ Make a least two velocity measurements that should agree with each other within less than 5% total difference. 6 Min-218 Lab #2 Air Quantity Measurement ❐ Make area measurements (Please see Sources of Error When Using Anemometer later in this chapter for further details). The intent of the anemometer traverse is to get the average velocity of the air moving through that place in the airway. This requires a movement of the anemometer to cover the section completely in the time selected and to cover an equal area in an equal time. This means keeping the same steady rate of travel and ending up exactly on time at the starting point. High-velocity Measurement Pitot tube is used in measuring high velocity and will be covered in future lab. 3 . Airway Area Measurements The most common method of determining the cross-sectional area of the air passage is through horizontal and vertical taped linear measurements. These may be reduced into average dimensions or parameters of regular shapes to produce an area. Alternatively the dimensions could be used to draw a plan of the section and the area determined through planimetry, weighing, or surveying computer software. There is a total of eight different ways recommended to measure an airway area2, with the vertical and horizontal tape measurement being the simplest and the most prevalent method underground. The airway is assumed to be of rectangular cross section and the area equals height times width. Several heights and widths (usually three: for heights, one at either side of the rib with the third one in the middle of the airway; for width, one near the roof, one in the middle, with the third reading taken close to the floor) are measured and averaged to come up with the representative area. 2 The eight recommended methods are: 1) Vertical and horizontal taping; 2) Vertical and horizontal offsets; 3) Simpson's rule; 4) Diagonal offset; 5) Spiked protractor; 6) Profilograph; 7) Full circle protractor; and 8) Photographic. 7 Min-218 Lab #2 Air Quantity Measurement Elementary Statistics 1 . Introduction We live in an age in which we are deluged by facts and figures, or "statistics," on almost every subject imaginable. Quantitative facts such as those indicated will form the basis of our study, but we are about to study more than just sets of data ==> inferential statistics: to use the data to make intelligent, rigorous statements (inferences) about a much larger phenomenon from which the data were selected. Ventilation data collection in the field certainly falls under this category. An understanding of some the basic statistical concepts will provide us with a tool that offers more accurate treatment of field data. Any problem in statistics has as its starting point a population of interest. 1) population is the total set of measurements of interest in a particular problem. The population measurements are unknown at the outset, and in many cases they cannot be completely determined. We cannot, for example, test all of factory A's components to get a complete list of lifelengths, for this would be too time-consuming and too expensive. We have to obtain a representative set of measurements from the population by performing an experiment. The set of measurements yielded by an experiment is called the sample. 2) A sample is a subset of the population that contains measurements obtained by an experiment. The objective is to use these sample data for purposes of making inferences about the population from which the sample was obtained. Questions as to the appropriate number and type of sample observations are important in any statistical investigation. A civil engineer wants to estimate the traffic intensity on a particular road. He might measure traffic flow by a mechanical device, such as the cables commonly seen stretched across a roadway. For how many days should he record the data to obtain a reliable estimate'? A mechanical engineer wants to test the strength of a certain material. The test is destructive and the material is expensive. Thus he wants to test as few specimens as possible. When can he stop testing and still make a reliable decision'? An environmental engineer wants to estimate the bacterial count in a lake. She will take samples from the lake and use a special culture to count the bacteria in the sample. How many samples does she need, and what should be the volume of water in each one'? 3) Deterministic Model To develop formally the methodology for inference making, we must start with a model for the phenomenon under study. A model may be thought of as a theoretical, and usually oversimplified, explanation of a complex system. In the sciences these models usually take the form of mathematical equation, such as the equations for heat transfer in physics or the equations for various chemical reactions. The models most familiar in the physical sciences and engineering are called deterministic models, which have the distinguishing feature that specific outcomes of experiments can be accurately predicted. 2 . Fundamental Statistical Techniques 1) Frequency Distribution – Histogram 8 Min-218 Lab #2 Air Quantity Measurement In order to estimate an estimate of a true result, a number of measurements must be made. Although the same thing is measured each time, there will be variations between the values obtained by successive measurements – called experimental error and always arises when measurements are made. (It does not mean that a mistake has been made in taking the readings. The variation could be due to a number of factors beyond the observer's control such as variation in the instruments, variation in the quantity being measured, etc.) ==> summarize to obtain useful information. Example 1 – Classroom temperature estimation (histogram, frequency distribution, etc.) Example 2 – Suppose twenty people were asked for their estimates of the correct time. These were the results obtained: 10:21 10:19 10:24 10:18 10:19 10:23 10:20 10:22 10:20 10:21 10:20 10:19 10:21 10:22 10:17 10:20 10:21 10:20 10:23 10:19 Value 10.17 ~ 10.18 10.19 ~ 10.20 10.21 ~ 10.22 10.23 ~ 10.24 Frequency 2 9 6 3 Cumulative Frequency 2 11 17 20 a) The boundaries of the histogram are midway between the class boundaries, 1 1 e.g., the rectangle representing 9 has boundaries 10:182 and 10:202; b) for the polygon, the frequencies are plotted at the mid-points of the class intervals. Histograms can be extremely instructive. They (1) show the range of values that can be expected (cumulative and otherwise); (2) can detect extreme values, possibly erroneous values; (3) the values that is occurring most often can be visually determined; (4) can also help to detect mixtures of geologic environments. Histogram diagram handout 2) Normal Distribution This is bell-shaped as shown below. It has a peak in the middle and tails off equally on both sides. The central value is the mean of the set of results. If the experimental variation is large, the results will be spread over a wide range; if the variation is small, the curve will have a narrow peak. Frequency Normal Distribution 9 Min-218 Lab #2 Air Quantity Measurement 3) Log-normal Distribution One pattern that occurs in almost dust measurements is the long-normal distribution curve. It has a long tail on the right and the peak is not at the center. Frequency Log-Normal Distribution ❐ Normal and lognormal distributions are two types commonly encountered in mineral deposit sampling. Usually, normal distribution will indicate that normal statistical parameters may be calculated for the data set. The mean (average) variance and standard deviation calculated for the sample data should provide a reasonable approximation of these parameters for the deposit. Lognormal distribution requires log parameters, geometric mean, and standard deviation. Various other types of distribution are also present. – Handout – mineral deposit distribution 100 Number of Samples Number of Samples 100 50 50 0 0 1.0 1.5 0 0 (a) Grade – % Cu 40 100 Number of Samples 100 Number of Samples 20 (b) Grade – % P2O5 50 50 0 0 0.5 (c) Grade – % Mo 1.0 0 0 10 20 (d) Grade – % O2 /T Ag (a) normal distribution, moderate variability, typical of some stratiform and massive sulfide deposits. (b) Normal distribution, low variability, found in certain industrial mineral, Fe, and Mn deposits. (c) Lognormal distribution, common in many Mo, Sn, W, and precious metal deposits. (d) Bimodal distribution which may be produced by sampling two distinct ore types, or sampling across a zonation boundary in the mineralization. ❐ In many type of deposits, the fact that sample data are not random is not of serious concern. The results may still be utilized while keeping in mind the bias introduced by the lack of statistical independence among the samples. 10 Min-218 Lab #2 Air Quantity Measurement 4) Best Estimate of the Correct Value – Mean Classical statistical analysis is based on two assumptions: (1) the samples are random and that (2) the data have a normal distribution. The best estimate of a correct for a set of data is the mean (X). sum of all observations Mean = Total number of observations taken x = 1n n ∑ i=1 xi = 1n (x1 + x2 + x3 + . . .x n ) But this is just a point estimate of the true mean (m) for the entire population, some measure of how close the point estimate is likely to be to the true value is required. One way to report is to report both the estimate and its standard deviation (standard error). 5) Dispersion – Standard Deviation (s) and variance (s) Once the correct value has been estimated, the extent of the scatter around this value must be estimated. A result will obviously be more reliable when the scatter is small than when it is large. Of course, the amount of scatter often depends on the magnitude of the result. For a normal and log-normal distribution (or transforming the data by taking logs to make it normal), the measure of the scatter may be given by the standard deviation, s, using the following formula (for small number of observations; n < 20): n 1 [ (xi - x)2 ] = n - 1 i∑ =1 σ= n 1 [ xi 2 - nx2 ] n - 1 i∑ =1 Variance, σ = s 2 In other words, the standard deviation can be thought of as the size of a "typical" deviation between an observed outcome and the expected value. Example 3 – Find the standard deviation of the following wet kata readings: 6.1 6.2 5.9 6.1 6.1 Mean, x = 6.16 Σx2 = 379.62 nx2 = 379.456 Σx2 – nx2 = 0.164 Σx2 – nx 2 n – 1 = 0.0182 σ= √ Σx2 – nx 2 n – 1 = 0.13 11 6.4 6.1 6.2 6.3 6.2 Min-218 Lab #2 Air Quantity Measurement The following formula is used to calculate standard deviation for cases there are large number of results: σ= n 1 [ x2i fi - nx2 ] n -1 i∑ =1 where xi is the mid-point of a small range, and ƒi is the frequency of observations in that range. 6) Limits – for observation data x x x x Temperature ❐ For a normally-distributed entire population, it is known that 66% of the readings will be within ± 1standard deviation of the mean; 95% will be within ± 2 standard deviations, and 99% will be within ± 3 standard deviations. ❐ This fact is used to put control limits on a set of results. For example, if 95% limits are required, it is known that 95 in 100 results will fall within these limits and 5 will fall outside. If more than 5 consistently fall outside, then these results are "out of control" and there have been changes in the experimental set-up, other than accounted for by experimental variation. For example, suppose a set of wet-bulb temperatures were taken in an airway. The mean and standard deviation of these results were calculated and the results were plotted on a graph as follows: x x x x x x x x x Mean xx x x Upper limit x x x x x Lower limit Time Once the limits have been established, 95% of the points should fall within them. If the graph looks like this: x x x x x x x x x Temperature x x x Upper limit x x x Mean x x x x 12 x x Lower limit x Min-218 Lab #2 Air Quantity Measurement then the temperature are higher than would be expected from normal experimental variation and the possible causes for this should be investigated. 7) Confidence intervals (CI) After knowing something about the mean (X) and variance (S) of θ as an estimator of q, it would be nice to know something about how small the distance between θ and q is likely to be. If θ is an estimator of q (which has a known sampling distribution), and one can find two quantities that depend on, say g1 (θ) and g2 (θ), such that P [ g1 (θ) ≤ q ≤ g2 (θ)] = 1 – a for some small positive number a, then we can say that (g1 (θ) and g2 (θ)) forms an interval that has a probability 1 – a of catpuring the true q. This interval is referred to as a confidence interval with confidence coefficient (1 – a), The quantity g1 (θ) is called the lower confidence limit (LCL) The quantity g2 (θ) is called the upper confidence limit (UCL) for some small positive number a, then we can say that (g1 (θ) and g2 (θ)) forms an interval that has a probability 1 – a of catpuring the true q. This interval is referred to as a confidence interval with confidence coefficient (1 – a), 8) Limits to a mean value – standard error (s) For estimating a mean m for a population with variance s2 , we select random sample X1 , X2 , . . . X n from this population and compute X as a point estimator of m. If n is large (n ≥ 30), then X has approximately a normal distribution with mean m and variance s2 /n. Formally, Z= X–µ σ/ n where Z is a variable which has a standard normal distribution, approximately. For any prescribed a we can find from Table a value Za/2 such that P[–z2/a ≤ Z ≤ + z2/a] = 1 – a Example 4 Data for 50 battery lifelength observations show that x = 2.266 and s = 1.935. Using the confidence interval x ± Zα/2 σ n 13 Min-218 Lab #2 Air Quantity Measurement with (1 – a) = 0.95, we see that ZZa/2 = Z0.025 = 1.96 (Table), this interval yields, 2.266 ± 1.96 1.935 , 2.266 ± 0.536 50 √ or (1.730, 2.802) Thus we are 95% confident (similarly formed intervals will contain m about 95% of the time in repeated sampling) that the true mean lies between 1.730 and 2.802. The term, (s/√ n), is called standard error where n is the number of readings taken. The constant 1.96 is usually rounded off to 2. From this we can state that: interval m ± 2s/√ n contains about 95% of the X that could be generated random samplings from the population. Thus in 95 cases out of 100, the true mean will lie within ±2 s n √ of the estimate. Same can be used to estimate ± 3 s s for 99% and ± for 66% CI. n √ n √ Example 5 Lab results (concentration of Pb) for a piece of lead reserve is as follows, 11.2 12.4 11.6 12.8 12.3 10.8 10.3 10.6 (n = 8) Mean, x = ΣnX = 11.5 Variance, S = Σ(X – X)2 S = (0.09 + 0.81 + 0.01 + 1.69 + 0.64 + 0.49 + 1.44 + 0.81)/7 = 5.978 The smallest value of S can assume is 0, and that would occur if all the probability was at a single point (X takes on a constant value with probability of 1). Standard deviation, σ = √ S = 2.445 The standard deviatiin can be thought of as the size of a "typical" deviation between an observed outcome and the expected value. 2.445 Standard error = σ = = 0.864 n √ √ 8 The 95% confidence interval that the concentration will fall within the range of: 14 Min-218 Lab #2 Air Quantity Measurement 11.5 ± 2 (standard error) or 11.5 – 2 (0.864) < Pb concentration < 11.5 + 2 (0.864) 9.771% < Pb concentration < 13.229% 9) Student Distribution (t-Distribution) If the samples we are dealing with are not large enough to ensure as approximately normal sampling distribution for X, same procedure is applied by still assuming normal distribution and arriving the same interval but using different constant – t-distribution, T= X–µ S/ n Since E(X) = µ, the sampling distribution of T should center at zero. Standard deviation (s) is replaced by s (random variable), the sampling distribution of T should show more variability than Z with (n – 1) degrees of freedom. Example 6 Prestressing wire for wrapping concrete pipe is manufactured in large rolls. A quality-control inspection required 5 specimens from a roll to be tested for ultimate tensile strength (UTS). The UTS measurements (1,000 psi) turned out to be 253, 261, 258, 255, 256. Use these data to construct a 95% CI estimate of the true mean UTS for the sampled roll. Is it assumed that if many wire specimens were tested, the relative frequency distribution of UTS measurements would be nearly normal. A CI based on the t distribution can then be employed. With 1 – a = 0.95 and n – 1 = 4 DF, t 0.025 = 2.776. From the observed data, x = 256.60 and s = 3.05 Thus x ± tα/2 s n 3.05 or 256.60 ± 3.79 5 √ We are 95% confident that the interval 252.81 to 260.39 includes the true mean UTS for the roll. becomes 256.60 ± (2.776) Example 7 (text, p. 476), 35 holes are drilled yielding a mean of 11.5% Pb, and the sampling s = 5.90; t-distribution for n = 35 (34 DF) at a 90% confidence interval is 1.691; the precision (one-half the range, r, of the confidence interval, is r t s (1.691)(5.90) = 1.69 (%) 2 = √ n = 35 √ confidence interval for the mean is 11.5% ± 1.69% Pb (or 11.5 ± 15%) 15 Min-218 Lab #2 Air Quantity Measurement If enough holes have to drilled to provide for a max 10% variation (±1.15% Pb rather than ±1.69% Pb) in the estimated grade at the 90% confidence level, t s 2 (1.691)(5.90) 2 n = [ r/2 ] = [ ] = 75.3 1.15 The appropriate values of t for the desired confidence level may be found in any handbook of statistical tables. At the 95% confidence level and greater than 50 samples, the value for t is approximately 2.0. Thus, if a particular set of 60 samples has a mean of 8.5% Pb with s = 1.2, then the confidence interval for the mean would be: CI = 8.5 ± 1.2 x 2 = 8.5 ± 0.31 60 √ CI = 8.19 < 8.5 < 8.81 There is only one chance in 20 (5% probability) that the true grade lies outside the range of 8.19% to 8.81% Pb. The formula can be rearranged to approximate the number of samples needed to reach a required precision for the estimate of the mean (p. 476), N= ((2)(s) CI/2 ) 2 For example, USBM requires that the grade determination should be within 20% of the estimated value in order to be qualified as proven ore. If the calculation of a set of sample data indicates an average grade of 1.0% Cu, then to be considered as proven, the grade must be between the confidence limits of 0.8% and 1.20% Cu. If s = 1.5, then, (2)(1.5) 2 N= = 225 0.4/2 ( ) If the preliminary sample consisted of 60 assays and produced a s of 1.5, then approximately 225 samples will be required before the grade of the deposit may be considered as meeting proven ore standards ==> Fig 2 offers a quick way for rapid approximations at 95% confidence interval. 16