document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1192439>Question 1192439</A>: <I><SPAN STYLE=\"word-break: break-all;\"> two independent samples of size n1 = 16 and n2 = 9 from two populations given       <BR>\n" );
document.write( "sum of X1  = 960 sum of x1 square + 38140, sum of X2 = 450 and sum of X2 square + 22700<BR>\n" );
document.write( "find 90% confidence interval for difference betweenn two population means<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 #848985 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=848985\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **1. Calculate Sample Means and Standard Deviations**\r<BR>\n" );
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document.write( "* **Sample 1:**<BR>\n" );
document.write( "    * Sample mean (x̄1) = sum(X1) / n1 = 960 / 16 = 60<BR>\n" );
document.write( "    * Sample variance (s1²) = (sum(x1²) - (sum(X1))² / n1) / (n1 - 1) <BR>\n" );
document.write( "                        = (38140 - (960)² / 16) / (16 - 1) <BR>\n" );
document.write( "                        = (38140 - 57600) / 15 <BR>\n" );
document.write( "                        = -129.33 <BR>\n" );
document.write( "    * Sample standard deviation (s1) = √s1² = √(-129.33) <BR>\n" );
document.write( "        * Note: Since the sample variance is negative, there might be an error in the provided data. \r<BR>\n" );
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document.write( "* **Sample 2:**<BR>\n" );
document.write( "    * Sample mean (x̄2) = sum(X2) / n2 = 450 / 9 = 50<BR>\n" );
document.write( "    * Sample variance (s2²) = (sum(X2²) - (sum(X2))² / n2) / (n2 - 1) <BR>\n" );
document.write( "                        = (22700 - (450)² / 9) / (9 - 1) <BR>\n" );
document.write( "                        = (22700 - 22500) / 8 <BR>\n" );
document.write( "                        = 25<BR>\n" );
document.write( "    * Sample standard deviation (s2) = √s2² = √25 = 5\r<BR>\n" );
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document.write( "**2. Calculate the Standard Error of the Difference**\r<BR>\n" );
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document.write( "* Standard Error (SE) = √[(s1²/n1) + (s2²/n2)] <BR>\n" );
document.write( "    * SE = √[(-129.33/16) + (25/9)] <BR>\n" );
document.write( "    * SE = √[-8.08 + 2.78] <BR>\n" );
document.write( "    * SE = √(-5.3) <BR>\n" );
document.write( "        * Note: The standard error is imaginary due to the negative sample variance in Sample 1. This indicates an issue with the provided data.\r<BR>\n" );
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document.write( "**3. Determine the Critical Value**\r<BR>\n" );
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document.write( "* For a 90% confidence interval, the critical value (Z-score) is 1.645.\r<BR>\n" );
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document.write( "**4. Calculate the Margin of Error**\r<BR>\n" );
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document.write( "* Margin of Error (ME) = Z-score * SE <BR>\n" );
document.write( "    * Since the standard error is imaginary, the margin of error cannot be calculated.\r<BR>\n" );
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document.write( "**5. Construct the Confidence Interval**\r<BR>\n" );
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document.write( "* Confidence Interval = (x̄1 - x̄2) ± ME <BR>\n" );
document.write( "    * Due to the issues with the data (negative sample variance), the confidence interval cannot be calculated.\r<BR>\n" );
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document.write( "**Conclusion**\r<BR>\n" );
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document.write( "* There appears to be an error in the provided data for Sample 1, as the calculated sample variance is negative. <BR>\n" );
document.write( "* Due to this error, the standard error and margin of error cannot be calculated, and therefore, the 90% confidence interval for the difference between the two population means cannot be determined.\r<BR>\n" );
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document.write( "**Recommendations**\r<BR>\n" );
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document.write( "* Double-check the provided data for Sample 1, especially the sum of squares of X1. <BR>\n" );
document.write( "* If the data is corrected, the calculations can be repeated to obtain the correct confidence interval.\r<BR>\n" );
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document.write( "**Important Note:** This analysis assumes that the two populations are normally distributed and independent.<BR>\n" );
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