1 PROPRIETARY MATERIAL. © The McGraw-Hill Companies, Inc. All rights reserved. No part of this Manual may be displayed, reproduced or distributed in any form or by any means, without the prior written permission of the publisher, or used beyond the limited distribution to teachers and educators permitted by McGraw-Hill for their individual course preparation. If you are a student using this Manual, you are using it without permission. CHAPTER 17 17.1 The data can be tabulated as i y (yi – y)2 1 8.8 0.7259 2 9.4 0.0635 3 10 0.1211 4 9.8 0.0219 5 10.1 0.2007 6 9.5 0.0231 7 10.1 0.2007 8 10.4 0.5595 9 9.5 0.0231 10 9.5 0.0231 11 9.8 0.0219 12 9.2 0.2043 13 7.9 3.0695 14 8.9 0.5655 15 9.6 0.0027 16 9.4 0.0635 17 11.3 2.7159 18 10.4 0.5595 19 8.8 0.7259 20 10.2 0.3003 21 10 0.1211 22 9.4 0.0635 23 9.8 0.0219 24 10.6 0.8987 25 8.9 0.5655 241.3 11.8624 (a) 241.39.65225y (b) 11.86240.70304125 1ys (c) 220.703041 0.494267ys (d) 0.703041c.v. 100% 7.28%9.652 (e) t0.05/2,25–1 = 2.063899 0.7030419.652 2.063899 9.36179925L 0.7030419.652 2.063899 9.94220125U (f) The data can be sorted and then grouped. We assume that if a number falls on the border between bins, it is placed in the lower bin. lower upper Frequency 7.5 8 1 8 8.5 0 8.5 9 4 9 9.5 7 9.5 10 62 PROPRIETARY MATERIAL. © The McGraw-Hill Companies, Inc. All rights reserved. No part of this Manual may be displayed, reproduced or distributed in any form or by any means, without the prior written permission of the publisher, or used beyond the limited distribution to teachers and educators permitted by McGraw-Hill for their individual course preparation. If you are a student using this Manual, you are using it without permission. 10 10.5 5 10.5 11 1 11 11.5 1 The histogram can then be constructed as 0123456787.75 8.25 8.75 8.5 8.5 8.5 7.5 7.833BinFrequency 17.2 The data can be sorted in ascending order and tabulated as i y (yi – y )2 1 28.15 2.829605 2 28.55 1.64389 3 28.65 1.397462 4 28.75 1.171033 5 29.05 0.611747 6 29.15 0.465319 7 29.25 0.33889 8 29.25 0.33889 9 29.35 0.232462 10 29.35 0.232462 11 29.45 0.146033 12 29.65 0.033176 13 29.65 0.033176 14 29.65 0.033176 15 29.65 0.033176 16 29.75 0.006747 17 29.75 0.006747 18 29.85 0.000319 19 30.15 0.101033 20 30.15 0.101033 21 30.25 0.174605 22 30.45 0.381747 23 30.45 0.381747 24 30.55 0.515319 25 30.65 0.66889 26 30.85 1.036033 27 31.25 2.010319 28 33.65 14.57603 835.3 29.50107 (a) 835.329.8321428y (b) median = 29.65 (c) mode = 29.65 (d) range = maximum – minimum = 33.65 – 28.15 = 5.53 PROPRIETARY MATERIAL. © The McGraw-Hill Companies, Inc. All rights reserved. No part of this Manual may be displayed, reproduced or distributed in any form or by any means, without the prior written permission of the publisher, or used beyond the limited distribution to teachers and educators permitted by McGraw-Hill for their individual course preparation. If you are a student using this Manual, you are using it without permission. (e) 29.501071.04529128 1ys (f) 221.045291 1.092632ys (g) 1.045291c.v. 100% 3.50%29.83214 (h) The data can be sorted and grouped. Lower Upper Midpoint Frequency28 28.4 28.2 1 28.4 28.8 28.6 3 28.8 29.2 29 2 29.2 29.6 29.4 5 29.6 30 29.8 7 30 30.4 30.2 3 30.4 30.8 30.6 4 30.8 31.2 31 1 31.2 31.6 31.4 1 31.6 32 31.8 32 32.4 32.2 32.4 32.8 32.6 32.8 33.2 33 33.2 33.6 33.4 33.6 34 33.8 1 The histogram can then be constructed as 01234567828.228.62929.429.830.230.63131.431.832.232.63333.433.8 (i) 68% of the readings should fall between ysy and ysy. That is, between 29.83214 – 1.04529 = 28.78685 and 29.83214 + 1.04529 = 30.87743. Twenty-two values fall between these bounds which is equal to 22/28 = 78.6% of the values which is somewhat higher than the expected value of 68% for the normal distribution. Here is a script showing how the problem would be answered using MATLAB’s built-in functions: clear,clf,clc,format compact y=[29.65 28.55 28.65 30.15 29.35 29.75 29.25 30.65 28.15 29.85 ... 29.05 30.25 30.85 28.75 29.65 30.45 29.15 30.45 33.65 29.35 ... 29.75 31.25 29.45 30.15 29.65 30.55 29.65 29.25]; meany=mean(y) mediany=median(y) modey=mode(y) range=max(y)-min(y) stddevy=std(y)4 PROPRIETARY MATERIAL. © The McGraw-Hill Companies, Inc. All rights reserved. No part of this Manual may be displayed, reproduced or distributed in any form or by any means, without the prior written permission of the publisher, or used beyond the limited distribution to teachers and educators permitted by McGraw-Hill for their individual course preparation. If you are a student using this Manual, you are using it without permission. variancey=var(y) cv=stddevy/meany ny=(34-28)/0.4-1; binmids=linspace(28+0.4,34-0.4,ny); hist(y,binmids) meany = 29.8321 mediany = 29.6500 modey = 29.6500 range = 5.5000 stddevy = 1.0453 variancey = 1.0926 cv = 0.0350 28 29 30 31 32 33 3401234567 17.3 The results can be summarized as y versus x x versus y Best fit equation y = 4.851535 + 0.35247x x = 9.96763 + 2.374101y Standard error 1.06501 2.764026 Correlation coefficient 0.914767 0.914767 We can also plot both lines on the same graph 048120 5 10 15 20yy versus xx versus yyx Thus, the “best” fit lines and the standard errors differ. This makes sense because different errors are being minimized depending on our choice of the dependent (ordinate) and independent (abscissa) variables. In contrast, the correlation coefficients are identical since the same amount of uncertainty is explained regardless of how the points are plotted.5 PROPRIETARY MATERIAL. © The McGraw-Hill Companies, Inc. All rights reserved. No part of this Manual may be displayed, reproduced or distributed in any form or by any means, without the prior written permission of the publisher, or used beyond the limited distribution to teachers and educators permitted by McGraw-Hill for their individual course preparation. If you are a student using this Manual, you are using it without permission. 17.4 The results can be summarized as /31.0589 0.78055 ( 4.476306; 0.901489)yxyxsr At x = 10, the best fit equation gives 23.2543. The line and data can be plotted along with the point (10, 10). 05101520253035010203040 The value of 10 is nearly 3 times the standard error away from the line, 23.2543 –
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