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Recognizing QR In None Using Barcode scanner for Software Control to read, scan read, scan image in Software applications. QR Code Creator In Visual C# Using Barcode drawer for .NET framework Control to generate, create QR image in .NET applications. 7.22. In Problem 7.21 would your conclusions be changed if it turned out that there was a significant difference in the mean grades of the classes Explain your answer. Drawing QR Code In .NET Framework Using Barcode generation for ASP.NET Control to generate, create QR Code image in ASP.NET applications. Painting Denso QR Bar Code In Visual Studio .NET Using Barcode creation for VS .NET Control to generate, create QR Code image in Visual Studio .NET applications. Since the actual mean grades were not used at all in Problem 7.21, it makes no difference what they are. This is to be expected in view of the fact that we are not attempting to decide whether there is a difference in mean grades, but only whether there is a difference in variability of the grades. QRCode Generation In Visual Basic .NET Using Barcode creation for .NET Control to generate, create QR Code 2d barcode image in .NET framework applications. Code 39 Printer In None Using Barcode drawer for Software Control to generate, create Code 39 image in Software applications. Operating characteristic curves 7.23. Referring to Problem 7.2, what is the probability of accepting the hypothesis that the coin is fair when the actual probability of heads is p 0.7 Draw Bar Code In None Using Barcode creator for Software Control to generate, create barcode image in Software applications. Bar Code Printer In None Using Barcode encoder for Software Control to generate, create bar code image in Software applications. The hypothesis H0 that the coin is fair, i.e., p 0.5, is accepted when the number of heads in 100 tosses lies between 39.5 and 60.5. The probability of rejecting H0 when it should be accepted (i.e., the probability of a Type I error) is represented by the total area a of the shaded region under the normal curve to the left in Fig. 76. As computed in Problem 7.2(a), this area a, which represents the level of significance of the test of H0, is equal to 0.0358. GTIN  12 Creation In None Using Barcode generation for Software Control to generate, create UPC Code image in Software applications. Generating Data Matrix In None Using Barcode encoder for Software Control to generate, create ECC200 image in Software applications. Fig. 76 Print NW7 In None Using Barcode printer for Software Control to generate, create ANSI/AIM Codabar image in Software applications. GTIN  128 Maker In .NET Framework Using Barcode printer for Reporting Service Control to generate, create EAN128 image in Reporting Service applications. If the probability of heads is p 0.7, then the distribution of heads in 100 tosses is represented by the normal curve to the right in Fig. 76. The probability of accepting H0 when actually p 0.7 (i.e., the probability of a Type II error) is given by the crosshatched area b. To compute this area, we observe that the distribution under the hypothesis p 0.7 has mean and standard deviation given by m np (100)(0.7) 70 s !npq 60.5 70 4.58 39.5 70 4.58 !(100)(0.7)(0.3) 2.07 6.66 4.58 Scanning UPC Symbol In None Using Barcode reader for Software Control to read, scan read, scan image in Software applications. Code 128 Code Set C Printer In None Using Barcode drawer for Office Excel Control to generate, create Code128 image in Microsoft Excel applications. 60.5 in standard units 39.5 in standard units
UPC  13 Generation In Java Using Barcode generation for Eclipse BIRT Control to generate, create EAN / UCC  13 image in BIRT applications. Code 128 Printer In None Using Barcode generator for Office Word Control to generate, create Code 128 image in Office Word applications. Then b area under the standard normal curve between z 6.66 and z 2.07 0.0192. Therefore, with the given decision rule there is very little chance of accepting the hypothesis that the coin is fair when actually p 0.7. Note that in this problem we were given the decision rule from which we computed a and b. In practice two other possibilities may arise: Create UPCA In C# Using Barcode generation for Visual Studio .NET Control to generate, create GS1  12 image in .NET framework applications. Reading Data Matrix ECC200 In C# Using Barcode scanner for .NET Control to read, scan read, scan image in .NET applications. CHAPTER 7 Tests of Hypotheses and Significance
(1) We decide on a (such as 0.05 or 0.01), arrive at a decision rule, and then compute b. (2) We decide on a and b and then arrive at a decision rule. 7.24. Work Problem 7.23 if (a) p
(a) If p
0.6, (b) p
0.8, (c) p
0.9, (d) p
0.6, the distribution of heads has mean and standard deviation given by m np (100)(0.6) 60 s !npq 60.5 60 4.90 39.5 60 4.90 !(100)(0.6)(0.4) 0.102 4.18 4.90 60.5 in standard units 39.5 in standard units
Then b area under the standard normal curve between z 4.18 and z 0.102 0.5405 Therefore, with the given decision rule there is a large chance of accepting the hypothesis that the coin is fair when actually p 0.6. (b) If p 0.8, then m np (100)(0.8) 80 and s !npq 60.5 4 39.5 4 80 80 !(100)(0.08)(0.2) 4.88 10.12 4.88 0.0000, very closely. 4. 60.5 in standard units 39.5 in standard units Then b
area under the standard curve between z
10.12 and z 0.9, b
(c) From comparison with (b) or by calculation, we see that if p (d) By symmetry, p 0.4 yields the same value of b as p 0 for all practical purposes. 0.5405. b) vs. p.
0.6, i.e., b
7.25. Represent the results of Problems 7.23 and 7.24 by constructing a graph of (a) b vs. p, (b) (1 Interpret the graphs obtained. Table 73 shows the values of b corresponding to given values of p as obtained in Problems 7.23 and 7.24. Note that b represents the probability of accepting the hypothesis p 0.5 when actually p is a value other than 0.5. However, if it is actually true that p 0.5, we can interpret b as the probability of accepting p 0.5 when it should be accepted. This probability equals 1 0.0358 0.9642 and has been entered into Table 73. Table 73 p b 0.1 0.0000 0.2 0.0000 0.3 0.0192 0.4 0.5405 0.5 0.9642 0.6 0.5405 0.7 0.0192 0.8 0.0000 0.9 0.0000 (a) The graph of b vs. p, shown in Fig. 77(a), is called the operating characteristic curve, or OC curve, of the decision rule or test of hypotheses. The distance from the maximum point of the OC curve to the line b 1 is equal to a 0.0358, the level of significance of the test. In general, the sharper the peak of the OC curve the better is the decision rule for rejecting hypotheses that are not valid. (b) The graph of (1 b) vs. p, shown in Fig. 77(b), is called the power curve of the decision rule or test of hypotheses. This curve is obtained simply by inverting the OC curve, so that actually both graphs are equivalent. The quantity 1 b is often called a power function since it indicates the ability or power of a test to reject hypotheses which are false, i.e., should be rejected. The quantity b is also called the operating characteristic function of a test.

