document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1193797>Question 1193797</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The table below gives the distribution of the number of hits by flying bombs in 450 equally sized areas in City X during World War II.<BR>\n" );
document.write( "Number of hit (x)  (0)	(1)	(2)	(3)	(4)	(5)	(6 or more)<BR>\n" );
document.write( "Frequency (f)	  (180)	(173)	(69)	(20)	(6)	(2)	(0)\r<BR>\n" );
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document.write( "Use χ^2  distribution and a 10% level of significance to test the adequacy of the Poisson distribution as a model for these data. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848594 by </B><A HREF=/tutors/aboutme.mpl?userid=proyaop><B>proyaop(69)</B></A><img src=\"/images/stars/stars-08.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=proyaop><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=848594\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **1. Calculate the Mean Number of Hits**\r<BR>\n" );
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document.write( "* Mean number of hits (λ) = (Σf * x) / Σf <BR>\n" );
document.write( "    * Where f is the frequency and x is the number of hits.<BR>\n" );
document.write( "    * λ = (0*180 + 1*173 + 2*69 + 3*20 + 4*6 + 5*2) / 450 <BR>\n" );
document.write( "    * λ = 0.5 \r<BR>\n" );
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document.write( "**2. Calculate Expected Frequencies under Poisson Distribution**\r<BR>\n" );
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document.write( "* Use the Poisson probability mass function: <BR>\n" );
document.write( "    * P(X = x) = (e^(-λ) * λ^x) / x! <BR>\n" );
document.write( "    * Where:<BR>\n" );
document.write( "        * X is the number of hits<BR>\n" );
document.write( "        * λ is the mean number of hits (0.5)<BR>\n" );
document.write( "        * e is the base of the natural logarithm (approximately 2.71828)<BR>\n" );
document.write( "        * x! is the factorial of x\r<BR>\n" );
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document.write( "* Calculate the expected frequency (E) for each number of hits:<BR>\n" );
document.write( "    * E(X = 0) = 450 * P(X = 0) = 450 * (e^(-0.5) * 0.5^0) / 0! = 270.27<BR>\n" );
document.write( "    * E(X = 1) = 450 * P(X = 1) = 450 * (e^(-0.5) * 0.5^1) / 1! = 135.14<BR>\n" );
document.write( "    * E(X = 2) = 450 * P(X = 2) = 450 * (e^(-0.5) * 0.5^2) / 2! = 33.78<BR>\n" );
document.write( "    * E(X = 3) = 450 * P(X = 3) = 450 * (e^(-0.5) * 0.5^3) / 3! = 5.63<BR>\n" );
document.write( "    * E(X = 4) = 450 * P(X = 4) = 450 * (e^(-0.5) * 0.5^4) / 4! = 0.70<BR>\n" );
document.write( "    * E(X = 5) = 450 * P(X = 5) = 450 * (e^(-0.5) * 0.5^5) / 5! = 0.07<BR>\n" );
document.write( "    * E(X ≥ 6) = 450 * (1 - [P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5)]) = 0.01\r<BR>\n" );
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document.write( "**3. Calculate the Chi-Square Test Statistic**\r<BR>\n" );
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document.write( "* **Chi-Square (χ²) = Σ [(O - E)² / E]** <BR>\n" );
document.write( "    * Where:<BR>\n" );
document.write( "        * O = Observed frequency<BR>\n" );
document.write( "        * E = Expected frequency\r<BR>\n" );
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document.write( "* **Calculate for each category:**<BR>\n" );
document.write( "    * (180 - 270.27)² / 270.27 = 32.18<BR>\n" );
document.write( "    * (173 - 135.14)² / 135.14 = 12.19<BR>\n" );
document.write( "    * (69 - 33.78)² / 33.78 = 36.03<BR>\n" );
document.write( "    * (20 - 5.63)² / 5.63 = 38.84<BR>\n" );
document.write( "    * (6 - 0.70)² / 0.70 = 42.57<BR>\n" );
document.write( "    * (2 - 0.07)² / 0.07 = 50.43<BR>\n" );
document.write( "    * (0 - 0.01)² / 0.01 = 0.01 \r<BR>\n" );
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document.write( "* **Sum the values:** χ² = 32.18 + 12.19 + 36.03 + 38.84 + 42.57 + 50.43 + 0.01 = 212.25\r<BR>\n" );
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document.write( "**4. Determine Degrees of Freedom**\r<BR>\n" );
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document.write( "* Degrees of Freedom (df) = Number of categories - Number of parameters estimated - 1<BR>\n" );
document.write( "    * Number of categories = 7 (0 hits, 1 hit, ..., 5 hits, 6 or more hits)<BR>\n" );
document.write( "    * Number of parameters estimated = 1 (the mean, λ)<BR>\n" );
document.write( "    * df = 7 - 1 - 1 = 5\r<BR>\n" );
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document.write( "**5. Find the Critical Value**\r<BR>\n" );
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document.write( "* Using a chi-square distribution table, find the critical value for α = 0.10 and df = 5. <BR>\n" );
document.write( "    * The critical value is approximately 9.24.\r<BR>\n" );
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document.write( "**6. Make a Decision**\r<BR>\n" );
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document.write( "* **Compare the calculated chi-square statistic to the critical value:**<BR>\n" );
document.write( "    * 212.25 > 9.24\r<BR>\n" );
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document.write( "* **Decision:** Since the calculated chi-square statistic (212.25) is greater than the critical value (9.24), we **reject the null hypothesis**.\r<BR>\n" );
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document.write( "**Conclusion**\r<BR>\n" );
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document.write( "* There is sufficient evidence at the 0.10 level of significance to conclude that the Poisson distribution is not an adequate fit for the observed data. <BR>\n" );
document.write( "* The observed frequencies of bomb hits deviate significantly from what would be expected under a Poisson distribution.\r<BR>\n" );
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document.write( "**Note:**\r<BR>\n" );
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document.write( "* This analysis assumes that the expected frequencies in each category are sufficiently large (generally, expected frequencies should be greater than 5). <BR>\n" );
document.write( "* For categories with expected frequencies less than 5, you may need to combine them with adjacent categories to meet this assumption.<BR>\n" );
document.write( " </SPAN>\n" );
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