document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1199549>Question 1199549</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A number of minor automobile accidents occur at various high-risk intersections in Teton County despite traffic lights. The Traffic Department claims that a modification in the type of light will reduce these accidents. The county commissioners have agreed to a proposed experiment. Eight intersections were chosen at random, and the lights at those intersections were modified. The numbers of minor accidents during a 6-month period before and after the modifications were:\r<BR>\n" );
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document.write( "After : 3  7  7 0 4 6 8 2 <BR>\n" );
document.write( "Before :  5 7 6 4 8 9 8 10\r<BR>\n" );
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document.write( "At the 0.01 significance level, is it reasonable to conclude that the modification reduced<BR>\n" );
document.write( "the number of traffic accidents?<BR>\n" );
document.write( "a. What are the null and alternate hypotheses?<BR>\n" );
document.write( "b. Compute the test statistic.<BR>\n" );
document.write( "c. Compute the p-value.<BR>\n" );
document.write( "d. What is your decision regarding the null hypothesis?<BR>\n" );
document.write( "e. Interpret the result.<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848193 by </B><A HREF=/tutors/aboutme.mpl?userid=textot><B>textot(100)</B></A><img src=\"/images/stars/stars-08.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=textot><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=848193\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **a. Hypotheses**\r<BR>\n" );
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document.write( "* **Null Hypothesis (H0):** The modification in the type of traffic light did not reduce the number of minor accidents. <BR>\n" );
document.write( "    * Mathematically: μ_before ≤ μ_after <BR>\n" );
document.write( "* **Alternative Hypothesis (H1):** The modification in the type of traffic light reduced the number of minor accidents.<BR>\n" );
document.write( "    * Mathematically: μ_before > μ_after \r<BR>\n" );
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document.write( "**b. Compute the Test Statistic**\r<BR>\n" );
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document.write( "* **Data:**<BR>\n" );
document.write( "    * After: 3, 7, 7, 0, 4, 6, 8, 2<BR>\n" );
document.write( "    * Before: 5, 7, 6, 4, 8, 9, 8, 10\r<BR>\n" );
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document.write( "* **Paired t-test:** Since we are comparing the same intersections before and after the modification, we use a paired t-test.\r<BR>\n" );
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document.write( "* **Calculate the differences:** <BR>\n" );
document.write( "    * 5 - 3 = 2<BR>\n" );
document.write( "    * 7 - 7 = 0<BR>\n" );
document.write( "    * 6 - 7 = -1<BR>\n" );
document.write( "    * 4 - 0 = 4<BR>\n" );
document.write( "    * 8 - 4 = 4<BR>\n" );
document.write( "    * 9 - 6 = 3<BR>\n" );
document.write( "    * 8 - 8 = 0<BR>\n" );
document.write( "    * 10 - 2 = 8\r<BR>\n" );
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document.write( "* **Calculate the mean and standard deviation of the differences:** <BR>\n" );
document.write( "    * Mean (d̄) = (2 + 0 - 1 + 4 + 4 + 3 + 0 + 8) / 8 = 2.625<BR>\n" );
document.write( "    * Standard deviation (s_d) = 2.878 (approximately)\r<BR>\n" );
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document.write( "* **Calculate the t-statistic:**<BR>\n" );
document.write( "    * t = (d̄ - 0) / (s_d / √n) <BR>\n" );
document.write( "    * t = (2.625 - 0) / (2.878 / √8) <BR>\n" );
document.write( "    * t ≈ 2.59\r<BR>\n" );
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document.write( "**c. Compute the p-value**\r<BR>\n" );
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document.write( "* **Degrees of freedom (df) = n - 1 = 8 - 1 = 7**<BR>\n" );
document.write( "* **Using a t-distribution table or statistical software:** Find the p-value associated with t = 2.59 and df = 7. <BR>\n" );
document.write( "    * **p-value ≈ 0.018**\r<BR>\n" );
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document.write( "**d. Decision Regarding the Null Hypothesis**\r<BR>\n" );
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document.write( "* **Significance level (α) = 0.01**<BR>\n" );
document.write( "* **Compare p-value to α:** <BR>\n" );
document.write( "    * p-value (0.018) > α (0.01)\r<BR>\n" );
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document.write( "* **Decision:** Since the p-value is greater than the significance level, we **fail to reject the null hypothesis**.\r<BR>\n" );
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document.write( "**e. Interpretation**\r<BR>\n" );
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document.write( "* **Conclusion:** At the 0.01 significance level, there is **not enough evidence** to conclude that the modification in the type of traffic light reduced the number of minor accidents. <BR>\n" );
document.write( "* **Explanation:** The observed difference in the number of accidents could be due to chance. \r<BR>\n" );
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document.write( "**Note:**\r<BR>\n" );
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document.write( "* This analysis assumes that the differences in the number of accidents are normally distributed.<BR>\n" );
document.write( "* A paired t-test is appropriate because we are comparing the same locations before and after the modification.<BR>\n" );
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