# 5 Use EXCEL and attach all spreadsheet analysis with solution) Fifty measurements of the ultimate tensile strength of wire are given in the accompanying table. a) Group the data and make an appropriate normalized histogram (with total area of histogram be 1 )to approximate the PDF. 2 b) Calculate ^ and ^ for the distribution from the ungrouped data. c) Using ^ and ^ from part b, draw a normal distribution through the normalized histogram .histogram. Ultimate Tensile Strength 103,779 102,325 102,325 103,799 102,906 104,651 105,377 100,145 104,796 105,087 104,796 103,799 103,197 106,395 106,831 103,488 100,872 100,872 105,087 102,906 97,383 104,360 103,633 101,017 101,162 101,453 107,848 104,651 98,110 103,779 99,563 103,197 104,651 101,162 105,813 105,337 102,906 102,470 108,430 101,744 103,633 105,232 106,540 106,104 102,616 106,831 101,744 100,726 103,924 101,598 Source: Data from E. B. Haugen, Probabilistic Mechanical Design Wiley, New York, 1980 (c) Determine the mean, median, and mode from the ungrouped data. (d) Determine the range and standard deviation from the ungrouped data (e) Plot the cumulative frequency distribution on normal-probability paper, and determine the mean and standard deviation. (f), for the data given in Table .what are the 95 percent confidence limits on the mean of the population? PROBLEM # 6 (Use EXCEL and attach all spreadsheet analysis with solution) Three sets of identical twenty five fatigue specimens were tested at the three different level of stresses.. The number of cycles to failure. The results are expressed as log N , were as follows. No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 TABLE 1: FATIGUE LIFE DATA NUMBER OF CYCLES TO FAILURES S1 S2 S3 380 MPa 340 MPa 300 MPa 34200 125500 954000 37700 156900 959400 42000 173600 1194600 42300 176900 1240500 48200 179400 1250400 52500 188500 1285500 55900 195100 1410500 58300 208100 1495100 61700 211900 1518700 64700 224100 1544700 65000 226000 1551400 65500 253000 1585900 70400 255500 1639100 71000 259000 1683700 72400 274000 1926100 75200 292000 2011300 77400 300400 2171800 77800 302300 2391500 87800 308300 2569400 93400 406300 2674900 94000 420700 2921700 97200 428500 3046500 99600 664800 3105500 116700 776100 3523200 122500 793900 4311700 Assume that the data at each stress level (S1, S2 and S3) is lognormally distributed. Using directly the data in table determine What is the mean fatigue life ( ) and its standard deviation ( )? What is the mean Ln of fatigue life ( lnN) and its standard deviation Ln of fatigue life (lnN)? What are the Parameters of Lognormal distributions at S1,S2and S3 PROBLEM # 7 A partially nished connecting rod is shown in the gure shown below. Each radius has a tolerance of 0.002. The tolerance of the distance L between the centers of the holes is 0.004. Find the tolerance for the dimension h, a b using 100% accurate dimensions (standard deviation is zero -an impossible ideal case), using a statistical view point of dimensions ,where each tolerance is . 3,and manufacturing process related to each dimension is centered on its nominal value