document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1195940>Question 1195940</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Dear tutor please help me with stats  questions, A marketing manager at a big retail shop claims that the variance in the weights of cereal boxes are less than 0.03 kilogram square. Thus, the weights (in kilograms) of a random sample of eight of these cereal boxes 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( "a) Test this clam at a 1% level of significance.\r<BR>\n" );
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document.write( "b) Estimate the variance of the entire population of cereal box weights with 90% confidence. </SPAN>\n" );
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document.write( "**1. Calculate Sample Statistics**\r<BR>\n" );
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document.write( "* **Sample Mean (x̄):** <BR>\n" );
document.write( "   * Calculate the average of the given weights. <BR>\n" );
document.write( "   * x̄ = (1.07 + 0.98 + 0.95 + 1.05 + 0.99 + 1.09 + 1.03 + 0.96) / 8 <BR>\n" );
document.write( "   * x̄ = 8.12 / 8 = 1.015 kg\r<BR>\n" );
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document.write( "* **Sample Variance (s²):**<BR>\n" );
document.write( "   * Calculate the variance of the sample using the formula:<BR>\n" );
document.write( "      * s² = Σ(x - x̄)² / (n - 1) <BR>\n" );
document.write( "      * where:<BR>\n" );
document.write( "         * x: Individual weight<BR>\n" );
document.write( "         * x̄: Sample mean<BR>\n" );
document.write( "         * n: Sample size (8)<BR>\n" );
document.write( "   * s² ≈ 0.0023 kg²\r<BR>\n" );
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document.write( "**2. State Hypotheses**\r<BR>\n" );
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document.write( "* **Null Hypothesis (H₀):** σ² = 0.03 (Variance of population weights is 0.03 kg²)<BR>\n" );
document.write( "* **Alternative Hypothesis (H₁):** σ² < 0.03 (Variance of population weights is less than 0.03 kg²)\r<BR>\n" );
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document.write( "**3. Determine 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 (0.0023)<BR>\n" );
document.write( "         * σ₀²: Hypothesized population variance (0.03)\r<BR>\n" );
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document.write( "   * χ² = (8 - 1) * 0.0023 / 0.03 <BR>\n" );
document.write( "   * χ² = 0.5333\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( "   * χ²_critical ≈ 2.167 (for a one-tailed test)\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 (0.5333) is less than the critical value (2.167), we **reject the null hypothesis**.\r<BR>\n" );
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document.write( "**Conclusion:**\r<BR>\n" );
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document.write( "* There is sufficient evidence at the 1% level of significance to support the claim that the variance in the weights of cereal boxes is less than 0.03 kg².\r<BR>\n" );
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document.write( "**b) 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 * 0.0023 / 14.067 ≈ 0.00114<BR>\n" );
document.write( "    * Upper Bound: (n - 1) * s² / χ²_lower = 7 * 0.0023 / 2.167 ≈ 0.0075\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 0.0011 kg² and 0.0075 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" );
document.write( "**a) Hypothesis Testing**\r<BR>\n" );
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document.write( "* **Hypotheses:**<BR>\n" );
document.write( "    * **Null Hypothesis (H0):** σ² ≥ 0.03 (Variance of cereal box weights is greater than or equal to 0.03 kg²)<BR>\n" );
document.write( "    * **Alternative Hypothesis (H1):** σ² < 0.03 (Variance of cereal box weights is less than 0.03 kg²)\r<BR>\n" );
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document.write( "* **Test Statistic:** <BR>\n" );
document.write( "    * We will use the 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<BR>\n" );
document.write( "            * σ₀² = hypothesized population variance (0.03)\r<BR>\n" );
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document.write( "* **Calculate Sample Variance (s²)**\r<BR>\n" );
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document.write( "1. Calculate the 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\r<BR>\n" );
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document.write( "2. 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\r<BR>\n" );
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document.write( "3. Sum the squared deviations.\r<BR>\n" );
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document.write( "4. Divide the sum of squared deviations by (n - 1) = 7 to get the sample variance (s²). <BR>\n" );
document.write( "   * s² ≈ 0.001486\r<BR>\n" );
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document.write( "* **Calculate Test Statistic:**\r<BR>\n" );
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document.write( "   * χ² = (8 - 1) * 0.001486 / 0.03 <BR>\n" );
document.write( "   * χ² ≈ 0.347\r<BR>\n" );
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document.write( "* **Determine Critical Value:**\r<BR>\n" );
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document.write( "   * Find the critical value of chi-square (χ²_critical) for a left-tailed test with α = 0.01 and degrees of freedom (df) = n - 1 = 7.<BR>\n" );
document.write( "   * Use a chi-square distribution table or a statistical software. <BR>\n" );
document.write( "   * For α = 0.01 and df = 7, χ²_critical ≈ 2.167\r<BR>\n" );
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document.write( "* **Decision Rule:**\r<BR>\n" );
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document.write( "   * If the calculated χ² is less than the critical value (χ² < χ²_critical), we fail to reject the null hypothesis.<BR>\n" );
document.write( "   * If the calculated χ² is greater than or equal to the critical value (χ² ≥ χ²_critical), we reject the null hypothesis.\r<BR>\n" );
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document.write( "* **Conclusion:**\r<BR>\n" );
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document.write( "   * Since 0.347 < 2.167, we fail to reject the null hypothesis. <BR>\n" );
document.write( "   * There is not enough evidence at the 1% significance level to support the claim that the variance in the weights of cereal boxes is less than 0.03 kg².\r<BR>\n" );
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document.write( "**b) Estimate the Variance with 90% Confidence**\r<BR>\n" );
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document.write( "* **Find Confidence Interval Bounds**\r<BR>\n" );
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document.write( "   * The confidence interval for the population variance is given by:\r<BR>\n" );
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document.write( "      * [(n - 1) * s² / χ²_upper, (n - 1) * s² / χ²_lower] \r<BR>\n" );
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document.write( "      * where:<BR>\n" );
document.write( "          * χ²_upper and χ²_lower are the upper and lower critical values of the chi-square distribution for (n - 1) degrees of freedom and the desired confidence level (90%).\r<BR>\n" );
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document.write( "* **Find Critical Values:**\r<BR>\n" );
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document.write( "   * For a 90% confidence level, α/2 = 0.05.<BR>\n" );
document.write( "   * Use a chi-square distribution table or software to find:<BR>\n" );
document.write( "      * χ²_upper (for α/2 = 0.05 and df = 7) <BR>\n" );
document.write( "      * χ²_lower (for 1 - α/2 = 0.95 and df = 7)\r<BR>\n" );
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document.write( "* **Calculate Confidence Interval:**<BR>\n" );
document.write( "   * Substitute the values of n, s², χ²_upper, and χ²_lower into the formula to calculate the lower and upper bounds of the confidence interval for the population variance.\r<BR>\n" );
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document.write( "**Note:**\r<BR>\n" );
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document.write( "* The specific values of χ²_upper and χ²_lower will depend on the chi-square distribution table or software used. <BR>\n" );
document.write( "* This analysis assumes that the weights of the cereal boxes are normally distributed.\r<BR>\n" );
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document.write( "This approach provides a method for estimating the population variance with 90% confidence.<BR>\n" );
document.write( " </SPAN>\n" );
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