document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1195912>Question 1195912</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A marketing manager at a big retail shop claims that the variance in the weights of cereal boxes are less<BR>\n" );
document.write( "than 0.003 kilogram<BR>\n" );
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document.write( ". Thus, the weights (in kilograms) of a random sample of eight of these cereal boxes\r<BR>\n" );
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document.write( "are listed here.\r<BR>\n" );
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document.write( "1.07 .98 .95 1.05 .99 1.09 1.03 .96\r<BR>\n" );
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document.write( "4.1 Test this clam at a 1 % level of significance (8 marks)<BR>\n" );
document.write( "4.2 Estimate the variance of the entire population of cereal box weights with 90% confidence.<BR>\n" );
document.write( "(7 marks) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848439 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=848439\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **4.1 Test this claim at a 1% level of significance**\r<BR>\n" );
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document.write( "**1. Hypotheses**\r<BR>\n" );
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document.write( "* **Null Hypothesis (H₀):** σ² ≥ 0.003 (Variance of cereal box weights is greater than or equal to 0.003 kg²)<BR>\n" );
document.write( "* **Alternative Hypothesis (H₁):** σ² < 0.003 (Variance of cereal box weights is less than 0.003 kg²)\r<BR>\n" );
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document.write( "**2. Calculate Sample Statistics**\r<BR>\n" );
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document.write( "* **Sample Mean (x̄):** <BR>\n" );
document.write( "    * x̄ = (1.07 + 0.98 + 0.95 + 1.05 + 0.99 + 1.09 + 1.03 + 0.96) / 8 = 1.015 kg\r<BR>\n" );
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document.write( "* **Sample Variance (s²):**<BR>\n" );
document.write( "    * Calculate the squared deviations from the mean: <BR>\n" );
document.write( "        * (1.07 - 1.015)² = 0.002925<BR>\n" );
document.write( "        * (0.98 - 1.015)² = 0.001225 <BR>\n" );
document.write( "        * ... and so on for all data points<BR>\n" );
document.write( "    * Sum the squared deviations.<BR>\n" );
document.write( "    * Divide the sum of squared deviations by (n - 1) = 7 to get the sample variance (s²) <BR>\n" );
document.write( "    * s² ≈ 0.001486 kg²\r<BR>\n" );
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document.write( "**3. Calculate Test Statistic**\r<BR>\n" );
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document.write( "* **Chi-Square Test Statistic:**<BR>\n" );
document.write( "    * χ² = (n - 1) * s² / σ₀² <BR>\n" );
document.write( "        * where:<BR>\n" );
document.write( "            * n: sample size (8)<BR>\n" );
document.write( "            * s²: sample variance (0.001486 kg²)<BR>\n" );
document.write( "            * σ₀²: hypothesized population variance (0.003 kg²)\r<BR>\n" );
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document.write( "    * χ² = (8 - 1) * 0.001486 / 0.003 <BR>\n" );
document.write( "    * χ² ≈ 3.46\r<BR>\n" );
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document.write( "**4. Determine Critical Value**\r<BR>\n" );
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document.write( "* **Degrees of Freedom:** df = n - 1 = 8 - 1 = 7<BR>\n" );
document.write( "* **Significance Level:** α = 0.01 (1%)<BR>\n" );
document.write( "* **Chi-Square Distribution Table:** <BR>\n" );
document.write( "    * Find the critical value (χ²_critical) from the chi-square distribution table with 7 degrees of freedom and α = 0.01. <BR>\n" );
document.write( "    * For a one-tailed test with α = 0.01 and df = 7, χ²_critical ≈ 2.167\r<BR>\n" );
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document.write( "**5. Decision Rule**\r<BR>\n" );
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document.write( "* **Reject H₀ if χ² < χ²_critical**\r<BR>\n" );
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document.write( "**6. Make a Decision**\r<BR>\n" );
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document.write( "* Since our calculated χ² (3.46) is greater than the critical value (2.167), we **fail to reject the null hypothesis**. \r<BR>\n" );
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document.write( "**7. Conclusion**\r<BR>\n" );
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document.write( "* There is not enough evidence at the 1% level of significance to support the claim that the variance in the weights of cereal boxes is less than 0.003 kg².\r<BR>\n" );
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document.write( "**4.2 Estimate the Variance of the Population with 90% Confidence**\r<BR>\n" );
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document.write( "* **Confidence Level:** 90%<BR>\n" );
document.write( "* **Degrees of Freedom:** df = n - 1 = 7<BR>\n" );
document.write( "* **Find Chi-Square Values:**<BR>\n" );
document.write( "    * Find the chi-square values (χ²_lower and χ²_upper) from the chi-square distribution table for 7 degrees of freedom and 5% and 95% significance levels (since it's a two-tailed interval).<BR>\n" );
document.write( "      * χ²_lower ≈ 2.167 <BR>\n" );
document.write( "      * χ²_upper ≈ 14.067\r<BR>\n" );
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document.write( "* **Calculate Confidence Interval:**<BR>\n" );
document.write( "    * Lower Bound: (n - 1) * s² / χ²_upper = 7 * 0.001486 / 14.067 ≈ 0.000738<BR>\n" );
document.write( "    * Upper Bound: (n - 1) * s² / χ²_lower = 7 * 0.001486 / 2.167 ≈ 0.004809\r<BR>\n" );
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document.write( "**Conclusion:**\r<BR>\n" );
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document.write( "* We are 90% confident that the true variance of the population of cereal box weights lies between 0.000738 kg² and 0.004809 kg².\r<BR>\n" );
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document.write( "**Note:**\r<BR>\n" );
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document.write( "* This analysis assumes that the weights of the cereal boxes are normally distributed.<BR>\n" );
document.write( "* The chi-square distribution is used to estimate the population variance.<BR>\n" );
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