Question

A. Introduction and Objective Every test has at least two sources of variation that affect the results of the test. The first source of variation is due to the experimental procedure, such as using two different testing machines that have different calibrations, or different observers reading the same equipment differently. This type of variation is often called the experimental error The second source of variation is inherent in the specimens (or sample population) themselves. In other words, no two specimens this variation can be very diffficult to determine, particularly if there is a large amount of experimental error. exactly alike. The exact amount of are In any experiment, it is important to quantify both types of variation and to determine if the experimental to address these issues in the conclusion of every lab report that you complete. error is more significant than the specimen variation. You are expected B. Theory 1.0 Distribution Plots A distribution plot shows the number of occurrences for some particular quantity. For example, a company that makes breakfast cereal may intend to fill each box with 500 grams of cereal. However, due to manufacturing variation, with more or less than 500 grams of cereal. some boxes are filled A certain number of cereal boxes can be tested, and then a distribution plot could be constructed based on the mass of the contents of each box of cereal. The plot has the mass on the x-axis and the number of occurrences for a particular range of mass on the y-axis (Figure 1) The type of distribution shown in Figure 1 is a discrete distribution of a finite sample population (size) because the x-axis (the mass) was broken down into 5 gram intervals and the number of boxes tested was 6180-a finite number. 2.0 Measures of Variance The average, deviance, variance, and standard deviation are used to describe variation in a sample population. For a finite population, the average or mean is given by -1j sample. The where n is the number of the population and x is the value of the deviation is the difference from the average for a certain sample: deviation- (x,-x) If all of the deviations are small, then there is a relatively small amount of variation in the sample population. The deviation for a particular member of the sample population can be positive or negative, depending on which side of the mean it lies. A physical interpretation of the variation of a particular value is the signed distance from the mean of a particular member of the sample population. The sample variance of a finite population, s, is given by ial n-1 The positive square root of the sample variance is the sample standard deviation. In the sample variance, the distance from the mean is squared to remove the negative sign. Extending the physical interpretation, the numerator in the expression is the summation of square areas whose sides are of the length given by the deviation for each particular member of the sample population. The denominator is one less than the total number of samples, because the sample population is finite. Two sample populations might have the same mean, but different values of the sample variance and standard deviation. This indicates that the dispersion of the distributions are different. 3.0 Continuous Distributions Often, these discrete distributions are represented by a continuous function, as in typically easier to work Figure 2. Continuous functions are useful because they are with than discrete distributions. 1200 Mass Distribution Cercal Box Contents 6180 Samples To Nearest 5 grams 1000 C. Crunch 23 July 1999 800 600 400 200 550 510 $30 470 490 450 M, Mass of Cereal (g) Figure 2. Discrete distribution represented by a continuous function Number of Occurrences at each Mass The continuous distribution shown in Figure 2 is a normal or Gaussian distribution The tails of the continuous distributions typically extend to plus and minus infinity Normal distributions have been demonstrated to be very useful in representing the distribution of a number of physical characteristics such as the variation of material yield strength, the distribution of measurement error, and even the distribution of exam grades in class. In the latter case, the distribution is often referred to as a bell- curve. Not all useful distributions can be represented by the normal distribution. In some cases, the data from the discrete population may be skewed to one direction or another. However, different types of continuous distributions can be used in these cases. Examples of different distributions are shown in Figure 3. (a) (b) (c) Value of Independent Variable Figure 3. Normal and skewed distributions. particular distribution can be prior experience to It is important to note that you should not assume a represented by the normal distribution unless there is evidence or support that assumption. You will learn more about probability distributions if you take a course in probability and statistics. Number of Occurrences D. Lab Report 1.0 Calculations Complete the columns in order in the table below to compute the sample mean, specimen variation, sample variance, and sample standard deviation. (You may fill in these columns, or print out similar columns using a spreadsheet program.) Be certain to use the appropriate number of significant digits. 13 Number of samples, n = (x- (x,-) (squared deviation) (crush load) (deviation from mean) 39.663 40.898 0.816 71.802 asal9 A4.489 $2.305 92.606 33.830 .253 78.253 g9.964 A0.553 36.229 lo3.533 36.5.34 Σ&-)- Σ&-3)- -Σx iel Average value (or mean value)- Σ4-) - (x Summation of deviations, Sample variance, s = Sample standard deviation, s= 2.0 Questions Answer the following in complete sentences. 1. What are some sources of experimental error, and what are some sources of sample population variation for the aluminum cans. What do you estimate to be the maximum experimental error based on the sources of experimental error listed above? 2. 3. Why is the sum of deviations zero or nearly zero? (Refer to column 2 of the table.) 4. Is the estimated experimental error greater than the sample standard deviation? What does it mean if the estimated experimental error is greater than the 5. sample standard deviation? 6. How do your results compare with the historical data? 3,0 Discussion of Results In every lab report that you complete, you must discuss the possible sources of variation and error in the experiment that was conducted. The discussion must include the estimated significance of the error (e.g. the load could only be determined to with +2.5 pounds). Reporting of results (including calculations) must be consistent with the proper number of significant digits attainable in the experiment. Write the discussion using complete sentences. Use additional pages if necessary

Answer 1

5.995296 8.786470588 77.20207 79.663 -2.448529412 1 2 90.898 90.81 .698470588 75.66339 -10.30952941 106.2864 13.60747059 185.1633 4 71.802 95.719 64.489 -17.62252941 310.5535 7 82.305 0.193470588 0.037431 92.606 10.49447059 110.1339 73.83 .281529412 68.58373 10 71.253 10.85852941 117.9077 11 78.253 3.858529412 14.88825 9,8841 -12.64452941 12 9,467 13 99.904 17.79247059 316.572 14 70.557 11.55452941 133.5071 3.117470588 9.718623 26.46547059 700.4211 15 85.229 16 108,577 17 70.534 -11.57752941 134.0392 2526.557 82.11153 0.00000000000025580 Average Value Summation of Deviation 82.11153 -0.000000000o002558 Sample Variance 157.9098 12.56622 Sample Standard Deviation Que 01:- As the data is varying on the both side of mean, hence it is a random error, this error get generated because of fluctuation in measurement during reading. Population variance might also be because in variation of can mechanical and geometric properties resulting in need of more crush load. Que 02-Our Crush load will vary between X - SD and X + SD, in this case between 69.545 and 94.677 Que 03:- Sum deviation is nearly zero becuase of Sum of Deviation above mean and Sum of deviation below mean always equal this is a property of Sample Statistics For Question no. 4, 5, 6 data is missing like I need theoretical crush load o estimated experimental error and also need historical data for comparision.