document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1207902>Question 1207902</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Independent random samples of n1 = 18 and n2 = 13 observations were selected from two normal populations with equal variances.<BR>\n" );
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document.write( "____________________Population   (Ignore the lines) <BR>\n" );
document.write( "____________________ 1_____2     (Ignore the lines)<BR>\n" );
document.write( "Sample Size_________18____13     (Ignore the lines)<BR>\n" );
document.write( "Sample Mean________34.6___32.1   (Ignore the lines)<BR>\n" );
document.write( "Sample Variance_____4.5___5.9    (Ignore the lines)\r<BR>\n" );
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document.write( "(a) Find the rejection region for the test in part (a) for 𝛼 = 0.01. (If the test is one-tailed, enter NONE for the unused region. Round your answers to three decimal places.)<BR>\n" );
document.write( "t > _______<BR>\n" );
document.write( "t < _______\r<BR>\n" );
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document.write( "(b) Find the value of the test statistic. (Round your answer to three decimal places.)<BR>\n" );
document.write( "t = \r<BR>\n" );
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document.write( "(c) Find the approximate p-value for the test.<BR>\n" );
document.write( "p-value < 0.010<BR>\n" );
document.write( "i) 0.010 < p-value < 0.020    <BR>\n" );
document.write( "ii) 0.020 < p-value < 0.050<BR>\n" );
document.write( "iii) 0.050 < p-value < 0.100<BR>\n" );
document.write( "iv) 0.100 < p-value < 0.200<BR>\n" );
document.write( "v) p-value < 0.200 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #847977 by </B><A HREF=/tutors/aboutme.mpl?userid=ElectricPavlov><B>ElectricPavlov(122)</B></A><img src=\"/images/stars/stars-09.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ElectricPavlov><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=847977\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **1. Calculate the Pooled Variance**\r<BR>\n" );
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document.write( "* Pooled variance (s_p^2) = ((n1 - 1) * s1^2 + (n2 - 1) * s2^2) / (n1 + n2 - 2)<BR>\n" );
document.write( "* s_p^2 = ((18 - 1) * 4.5 + (13 - 1) * 5.9) / (18 + 13 - 2) <BR>\n" );
document.write( "* s_p^2 = 5.06\r<BR>\n" );
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document.write( "**2. Calculate the Degrees of Freedom**\r<BR>\n" );
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document.write( "* Degrees of freedom (df) = n1 + n2 - 2 = 18 + 13 - 2 = 29\r<BR>\n" );
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document.write( "**3. Find the Critical Value**\r<BR>\n" );
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document.write( "* For a two-tailed test with α = 0.01 and df = 29, the critical values are:<BR>\n" );
document.write( "    * t_critical = ±2.756 \r<BR>\n" );
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document.write( "**4. Calculate the Test Statistic**\r<BR>\n" );
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document.write( "* t = (x̄1 - x̄2) / √(s_p^2 * (1/n1 + 1/n2))<BR>\n" );
document.write( "* t = (34.6 - 32.1) / √(5.06 * (1/18 + 1/13))<BR>\n" );
document.write( "* t = 3.048\r<BR>\n" );
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document.write( "**5. Determine the P-value**\r<BR>\n" );
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document.write( "* Using a t-distribution table or statistical software, find the p-value associated with the calculated t-statistic (3.048) and degrees of freedom (29).<BR>\n" );
document.write( "* The p-value will be less than 0.01.\r<BR>\n" );
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document.write( "**Therefore:**\r<BR>\n" );
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document.write( "* **(a) Rejection Region:**<BR>\n" );
document.write( "    * t > 2.756<BR>\n" );
document.write( "    * t < -2.756\r<BR>\n" );
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document.write( "* **(b) Test Statistic (t): 3.048**\r<BR>\n" );
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document.write( "* **(c) P-value: p-value < 0.010**<BR>\n" );
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