document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1193539>Question 1193539</A>: <I><SPAN STYLE=\"word-break: break-all;\"> (1) The weight of cement bags produced in a cement company follow normal distribution <BR>\n" );
document.write( "whose population is infinite. The expected mean of weight of the cement bags for sales of <BR>\n" );
document.write( "this population is 45KG and its variance is unknown. The sales manager of the firm <BR>\n" );
document.write( "claims that the mean weight of the cement bags is significantly more than the expected <BR>\n" );
document.write( "mean weight of the population. So, he has selected a random sample of 36 bags and its <BR>\n" );
document.write( "mean and variance are found to be 44KG and 1.25KG, respectively. Verify the intuition <BR>\n" );
document.write( "of the sales manager at a significance level of 0.05. \r<BR>\n" );
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document.write( "(2) The quality manager of a washing machine company feels that the mean time between <BR>\n" );
document.write( "failures of the motors received is at most 90 days. The quality manager wants to test his <BR>\n" );
document.write( "intuition. Hence, he has taken a sample of 25 motors whose mean time between failure <BR>\n" );
document.write( "and its variance are found to be 93 days and 16 days, respectively. Verify the intuition of <BR>\n" );
document.write( "the quality manager 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 #848650 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=848650\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Certainly, let's analyze both scenarios.\r<BR>\n" );
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document.write( "**1. Cement Bag Weights**\r<BR>\n" );
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document.write( "**a) Set up Hypotheses**\r<BR>\n" );
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document.write( "* **Null Hypothesis (H0):** The mean weight of the cement bags is less than or equal to the expected mean weight. <BR>\n" );
document.write( "    * μ ≤ 45 kg <BR>\n" );
document.write( "* **Alternative Hypothesis (H1):** The mean weight of the cement bags is significantly greater than the expected mean weight.<BR>\n" );
document.write( "    * μ > 45 kg\r<BR>\n" );
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document.write( "**b) Choose the Test Statistic**\r<BR>\n" );
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document.write( "* Since the population variance is unknown and the sample size is greater than 30, we can use the **one-sample t-test**.\r<BR>\n" );
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document.write( "**c) Calculate the Test Statistic**\r<BR>\n" );
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document.write( "* **Given:**<BR>\n" );
document.write( "    * Sample size (n): 36<BR>\n" );
document.write( "    * Sample mean (x̄): 44 kg<BR>\n" );
document.write( "    * Sample variance (s²): 1.25 kg²<BR>\n" );
document.write( "    * Population mean (μ₀): 45 kg<BR>\n" );
document.write( "* **Calculate the standard error:**<BR>\n" );
document.write( "    * Standard Error (SE) = s / √n <BR>\n" );
document.write( "        * SE = √(1.25) / √36 <BR>\n" );
document.write( "        * SE ≈ 0.2041 <BR>\n" );
document.write( "* **Calculate the t-score:**<BR>\n" );
document.write( "    * t = (x̄ - μ₀) / SE <BR>\n" );
document.write( "        * t = (44 - 45) / 0.2041 <BR>\n" );
document.write( "        * t ≈ -4.898\r<BR>\n" );
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document.write( "**d) Determine Critical Value**\r<BR>\n" );
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document.write( "* **Significance Level:** α = 0.05<BR>\n" );
document.write( "* **Degrees of Freedom (df):** n - 1 = 36 - 1 = 35<BR>\n" );
document.write( "* **One-tailed test (right-tailed):** Find the critical t-value from a t-distribution table. <BR>\n" );
document.write( "    * t_critical ≈ 1.690\r<BR>\n" );
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document.write( "**e) Decision Rule**\r<BR>\n" );
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document.write( "* If the calculated t-score is greater than the critical t-value, reject the null hypothesis.<BR>\n" );
document.write( "* If the calculated t-score is less than or equal to the critical t-value, fail to reject the null hypothesis.\r<BR>\n" );
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document.write( "**f) Make a Decision**\r<BR>\n" );
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document.write( "* Our calculated t-score (-4.898) is less than the critical t-value (1.690).\r<BR>\n" );
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document.write( "* **Conclusion:** We fail to 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 does not support the sales manager's claim that the mean weight of the cement bags is significantly greater than the expected mean weight.\r<BR>\n" );
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document.write( "**2. Washing Machine Motor Time Between Failures**\r<BR>\n" );
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document.write( "**a) Set up Hypotheses**\r<BR>\n" );
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document.write( "* **Null Hypothesis (H0):** The mean time between failures is less than or equal to 90 days.<BR>\n" );
document.write( "    * μ ≤ 90 days <BR>\n" );
document.write( "* **Alternative Hypothesis (H1):** The mean time between failures is greater than 90 days.<BR>\n" );
document.write( "    * μ > 90 days\r<BR>\n" );
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document.write( "**b) Choose the Test Statistic**\r<BR>\n" );
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document.write( "* Similar to the first scenario, we can use the **one-sample t-test** since the population variance is unknown.\r<BR>\n" );
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document.write( "**c) Calculate the Test Statistic**\r<BR>\n" );
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document.write( "* **Given:**<BR>\n" );
document.write( "    * Sample size (n): 25<BR>\n" );
document.write( "    * Sample mean (x̄): 93 days<BR>\n" );
document.write( "    * Sample variance (s²): 16 days²<BR>\n" );
document.write( "    * Population mean (μ₀): 90 days<BR>\n" );
document.write( "* **Calculate the standard error:**<BR>\n" );
document.write( "    * SE = s / √n <BR>\n" );
document.write( "        * SE = √(16) / √25 <BR>\n" );
document.write( "        * SE = 4 / 5 <BR>\n" );
document.write( "        * SE = 0.8 days<BR>\n" );
document.write( "* **Calculate the t-score:**<BR>\n" );
document.write( "    * t = (x̄ - μ₀) / SE <BR>\n" );
document.write( "        * t = (93 - 90) / 0.8 <BR>\n" );
document.write( "        * t = 3 / 0.8 <BR>\n" );
document.write( "        * t = 3.75\r<BR>\n" );
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document.write( "**d) Determine Critical Value**\r<BR>\n" );
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document.write( "* **Significance Level:** α = 0.05<BR>\n" );
document.write( "* **Degrees of Freedom (df):** n - 1 = 25 - 1 = 24<BR>\n" );
document.write( "* **One-tailed test (right-tailed):** Find the critical t-value from a t-distribution table. <BR>\n" );
document.write( "    * t_critical ≈ 1.711\r<BR>\n" );
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document.write( "**e) Decision Rule**\r<BR>\n" );
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document.write( "* If the calculated t-score is greater than the critical t-value, reject the null hypothesis.<BR>\n" );
document.write( "* If the calculated t-score is less than or equal to the critical t-value, fail to reject the null hypothesis.\r<BR>\n" );
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document.write( "**f) Make a Decision**\r<BR>\n" );
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document.write( "* Our calculated t-score (3.75) is greater than the critical t-value (1.711).\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 supports the quality manager's intuition that the mean time between failures of the motors is significantly greater than 90 days.<BR>\n" );
document.write( " </SPAN>\n" );
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