document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1207925>Question 1207925</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 - represents space)<BR>\n" );
document.write( "__________________1______2     (IGNORE the lines - represents space)<BR>\n" );
document.write( "Sample Size______18______13  (IGNORE the lines - represents space)<BR>\n" );
document.write( "Sample Mean_____34.6_____32.1 (IGNORE the lines - represents space)<BR>\n" );
document.write( "Sample Variance__4.5_____5.9   (IGNORE the lines - represents space)\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( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848147 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=848147\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **a) Find the rejection region for the test in part (a) for 𝛼 = 0.01.**\r<BR>\n" );
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document.write( "* **Degrees of Freedom:**<BR>\n" );
document.write( "  * df = n1 + n2 - 2 = 18 + 13 - 2 = 29\r<BR>\n" );
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document.write( "* **Critical t-values:**<BR>\n" );
document.write( "  * Since it's a two-tailed test with α = 0.01, we need to find the critical t-values that split the distribution into the middle 98% and the two 1% tails.<BR>\n" );
document.write( "  * Using a t-distribution table or a calculator (like Python's scipy library), we find:<BR>\n" );
document.write( "    * t > 2.756 <BR>\n" );
document.write( "    * t < -2.756\r<BR>\n" );
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document.write( "**b) Find the value of the test statistic.**\r<BR>\n" );
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document.write( "1. **Calculate Pooled Variance:**<BR>\n" );
document.write( "   * s_pooled^2 = ((n1 - 1) * s1^2 + (n2 - 1) * s2^2) / (n1 + n2 - 2)<BR>\n" );
document.write( "   * s_pooled^2 = ((18 - 1) * 4.5 + (13 - 1) * 5.9) / (18 + 13 - 2) <BR>\n" );
document.write( "   * s_pooled^2 = 5.14 <BR>\n" );
document.write( "   * s_pooled = √5.14 ≈ 2.267\r<BR>\n" );
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document.write( "2. **Calculate t-statistic:**<BR>\n" );
document.write( "   * t = (x̄1 - x̄2) / (s_pooled * √(1/n1 + 1/n2))<BR>\n" );
document.write( "   * t = (34.6 - 32.1) / (2.267 * √(1/18 + 1/13))<BR>\n" );
document.write( "   * t ≈ 3.048\r<BR>\n" );
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document.write( "**Therefore:**\r<BR>\n" );
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document.write( "* **Rejection Region:** t > 2.756 and t < -2.756<BR>\n" );
document.write( "* **Test Statistic:** t = 3.048<BR>\n" );
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