document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1193999>Question 1193999</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A lab technician is tested for her consistency by taking multiple measurements of cholesterol levels from the same blood sample. The target accuracy is a standard deviation less than 1.2. If 20 measurements are taken and their standard deviation is 1.1, is there enough evidence at a=.05 that she is meeting the target accuracy?\r<BR>\n" );
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document.write( "The correct Null and Alternative Hypothesis statement would be?\r<BR>\n" );
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document.write( "The correct critical value would be?\r<BR>\n" );
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document.write( "The correct test value would be?\r<BR>\n" );
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document.write( "The decision and summary statement would be? <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 #848535 by </B><A HREF=/tutors/aboutme.mpl?userid=parmen><B>parmen(42)</B></A><img src=\"/images/stars/stars-07.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=parmen><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=848535\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **1. Set up Hypotheses**\r<BR>\n" );
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document.write( "* **Null Hypothesis (H₀):** σ² ≥ 1.2² (The population variance of the technician's measurements is greater than or equal to 1.44) <BR>\n" );
document.write( "* **Alternative Hypothesis (H₁):** σ² < 1.2² (The population variance of the technician's measurements is less than 1.44)\r<BR>\n" );
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document.write( "**2. Determine the Test Statistic**\r<BR>\n" );
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document.write( "* We will use the Chi-Square test statistic for testing the population variance.<BR>\n" );
document.write( "* **Test Statistic:** χ² = (n - 1) * s² / σ₀² <BR>\n" );
document.write( "    * where:<BR>\n" );
document.write( "        * n = sample size (20 measurements)<BR>\n" );
document.write( "        * s² = sample variance (1.1² = 1.21)<BR>\n" );
document.write( "        * σ₀² = hypothesized population variance (1.2² = 1.44)\r<BR>\n" );
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document.write( "* **Calculate the Test Statistic:**<BR>\n" );
document.write( "    * χ² = (20 - 1) * 1.21 / 1.44 = 19 * 1.21 / 1.44 = 15.96\r<BR>\n" );
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document.write( "**3. Determine the Critical Value**\r<BR>\n" );
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document.write( "* **Degrees of Freedom (df):** df = n - 1 = 20 - 1 = 19<BR>\n" );
document.write( "* **Significance Level (α):** α = 0.05 <BR>\n" );
document.write( "* **Critical Value:** <BR>\n" );
document.write( "    * Since this is a left-tailed test, we find the critical value from the Chi-Square distribution table with 19 degrees of freedom and an area of 0.05 in the left tail. <BR>\n" );
document.write( "    * Using a Chi-Square table or statistical software, the critical value for χ²(0.05, 19) is approximately 8.907.\r<BR>\n" );
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document.write( "**4. Decision**\r<BR>\n" );
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document.write( "* **Compare Test Statistic to Critical Value:**<BR>\n" );
document.write( "    * Our calculated test statistic (15.96) is greater than the critical value (8.907).\r<BR>\n" );
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document.write( "* **Decision:** <BR>\n" );
document.write( "    * Since the test statistic does not fall in the rejection region (less than the critical value), we **fail to reject the null hypothesis**.\r<BR>\n" );
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document.write( "**5. Summary Statement**\r<BR>\n" );
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document.write( "* There is **not enough evidence** at the 0.05 significance level to conclude that the lab technician is meeting the target accuracy of a standard deviation less than 1.2. \r<BR>\n" );
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document.write( "**Key Points:**\r<BR>\n" );
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document.write( "* We use the Chi-Square test for testing hypotheses about population variance.<BR>\n" );
document.write( "* The critical value is determined based on the significance level and degrees of freedom.<BR>\n" );
document.write( "* We compare the test statistic to the critical value to make a decision about the null hypothesis.<BR>\n" );
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