document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1193541>Question 1193541</A>: <I><SPAN STYLE=\"word-break: break-all;\">  The weight of a drug produced by a pharmaceutical company follows normal <BR>\n" );
document.write( "distribution. The specified variances of the weight of the drug of this population is <BR>\n" );
document.write( "0.36KG. The quality engineer of the firm claims that the variance of the weight of the <BR>\n" );
document.write( "drug does not differ significantly from the specified variance of the weight of the drug of <BR>\n" );
document.write( "the population. So, the purchase manager of a hospital who places order for the drug with <BR>\n" );
document.write( "the pharmaceutical company has selected a random sample of 15 drugs. The variance of <BR>\n" );
document.write( "the weight of the sample is found to be 0.05KG. Verify the intuition of the quality <BR>\n" );
document.write( "engineer of the pharmaceutical company at a significance level of 0.05. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848648 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=848648\">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):** The variance of the drug weight in the sample is equal to the specified population variance. <BR>\n" );
document.write( "    * σ² = 0.36 kg² \r<BR>\n" );
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document.write( "* **Alternative Hypothesis (H1):** The variance of the drug weight in the sample is different from the specified population variance.<BR>\n" );
document.write( "    * σ² ≠ 0.36 kg²\r<BR>\n" );
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document.write( "**2. Test Statistic**\r<BR>\n" );
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document.write( "* We will use the chi-square test statistic for this hypothesis test:\r<BR>\n" );
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document.write( "   χ² = (n - 1) * s² / σ² \r<BR>\n" );
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document.write( "   where:<BR>\n" );
document.write( "     * n is the sample size (15)<BR>\n" );
document.write( "     * s² is the sample variance (0.05 kg²)<BR>\n" );
document.write( "     * σ² is the population variance (0.36 kg²)\r<BR>\n" );
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document.write( "**3. Calculate Test Statistic**\r<BR>\n" );
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document.write( "χ² = (15 - 1) * 0.05 / 0.36 <BR>\n" );
document.write( "χ² = 14 * 0.05 / 0.36<BR>\n" );
document.write( "χ² ≈ 1.94\r<BR>\n" );
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document.write( "**4. Determine Critical Values**\r<BR>\n" );
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document.write( "* **Degrees of Freedom:** df = n - 1 = 15 - 1 = 14<BR>\n" );
document.write( "* **Significance Level:** α = 0.05<BR>\n" );
document.write( "* **Two-tailed test:** We need to find the critical values for both tails of the chi-square distribution.\r<BR>\n" );
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document.write( "* **Find critical values using a chi-square table or statistical software:**<BR>\n" );
document.write( "    * Lower critical value (χ²_lower) ≈ 5.629<BR>\n" );
document.write( "    * Upper critical value (χ²_upper) ≈ 26.119\r<BR>\n" );
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document.write( "**5. Decision Rule**\r<BR>\n" );
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document.write( "* If the calculated chi-square statistic (χ²) falls within the critical region (below χ²_lower or above χ²_upper), we reject the null hypothesis.<BR>\n" );
document.write( "* If the calculated chi-square statistic falls within the acceptance region (between χ²_lower and χ²_upper), we fail to reject the null hypothesis.\r<BR>\n" );
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document.write( "**6. Make a Decision**\r<BR>\n" );
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document.write( "* Our calculated χ² (1.94) is less than the lower critical value (5.629). \r<BR>\n" );
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document.write( "* **Conclusion:** We reject the null hypothesis. \r<BR>\n" );
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document.write( "**Interpretation**\r<BR>\n" );
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document.write( "The evidence suggests that the variance of the drug weight in the sample is significantly different from the specified population variance at the 0.05 significance level. \r<BR>\n" );
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document.write( "**Therefore, the quality engineer's claim that the variance of the drug weight does not differ significantly from the specified variance is not supported by the sample data.**<BR>\n" );
document.write( " </SPAN>\n" );
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