Intro to Probabiltiy theory notes for Elements Class.pptx

# When x 120kg z 1 120 15115 207 when x 155kg z 2 155

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When x = 120Kg z 1 = (120 – 151)/15 = -2.07 When x = 155Kg z 2 = (155 – 151)/15 = 0.27 Probability

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111 Pr mass between 120Kg and 155Kg is the sum of the areas to the left and the right of the centre. Area = 0.4808 + 0.1064 Area = Pr = 0.5872 (b) To find Pr mass >185Kg z = (185 – 151)/15 = 2.27 Pr of having a mass >185 is the shaded area. This is 0.5 – z 2.27 Area = 0.5 – 0.4884 Area = 0.0116 Probability
112 Practical Problems on the Normal Distribution 2. The lengths of rods held in stock have a mean of 4.5 m with a standard deviation of 0.05. The distribution of lengths is normal. If the total stock is 1000 rods, how many of the rods can be expected to have (a) a length between 4.5 and 4.6 m and (b) a length less than 4.4 m? Note The number of rods = Pr x total number of rods. (a)μ = 4.5 m and σ = 0.05 m – we need to calculate z for each value at the extremes of the range. When x = 4.5 m z 1 = (4.5 – 4.5)/0.05 = 0.00 When x = 4.6 m z 2 = (4.6 – 4.5)/0.05 = 2.00 The probability of the rod length being between 4.5 and 4.6 m is shown in the diagram on the next

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113 Pr = Area = 0.477.2 Number of rods equals 0.4772 x 1000 = 477 rods (b)To find the number of rods < 4.4 m z = (4.4 – 4.5)/0.05 = -2.00 Pr of having a length < 4.4 is the shaded area. This is 0.5 – z -2.00 Area = 0.5 – 0.4772 Area = 0.0228 = 23 rods
114 Examples 1. The mean braking load of a sample of concrete blocks was found to be 600N with a standard deviation of 45N. If the loads are normally distributed, calculate the Pr that another block would have a braking load of (a)between 585 and 610 N [0.2164] (b)less than 540N [0.0918] 2. The mean percentage moisture content of a series of 500 test samples of a particular cement mix after a certain time was found to be 27.5% with a standard deviation of 0.5%. Assuming the distribution is normal, find how many of the test samples have a moisture content below 27.1% [106] Probability

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115 This resource was created by the University of Wales Newport and released as an open educational resource through the Open Engineering Resources project of the HE Academy Engineering Subject Centre. The Open Engineering Resources project was funded by HEFCE and part of the JISC/HE Academy UKOER programme. © 2009 University of Wales Newport This work is licensed under a Creative Commons Attribution 2.0 License . The JISC logo is licensed under the terms of the Creative Commons Attribution-Non-Commercial-No Derivative Works 2.0 UK: England & Wales Licence. All reproductions must comply with the terms of that licence. The HEA logo is owned by the Higher Education Academy Limited may be freely distributed and copied for educational purposes only, provided that appropriate acknowledgement is given to the Higher Education Academy as the copyright holder and original publisher. The name and logo of University of Wales Newport is a trade mark and all rights in it are reserved. The name and logo should not be reproduced without the express authorisation of the University. Probability
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