document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1192645>Question 1192645</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Assignment 3<BR>\n" );
document.write( "1. Wick Ltd. manufactures washers. One of the key quality measures is the thickness of the washers. A sample of 5 washers in 7 different groups were recorded as shown in the table.<BR>\n" );
document.write( " <BR>\n" );
document.write( "Thickness (millimetres)<BR>\n" );
document.write( " <BR>\n" );
document.write( "1<BR>\n" );
document.write( "2<BR>\n" );
document.write( "3<BR>\n" );
document.write( "4<BR>\n" );
document.write( "5<BR>\n" );
document.write( "1<BR>\n" );
document.write( "2.8<BR>\n" );
document.write( "3.3<BR>\n" );
document.write( "3.0<BR>\n" );
document.write( "3.1<BR>\n" );
document.write( "2.9<BR>\n" );
document.write( "2<BR>\n" );
document.write( "3.5<BR>\n" );
document.write( "3.4<BR>\n" );
document.write( "3.2<BR>\n" );
document.write( "2.9<BR>\n" );
document.write( "3.3<BR>\n" );
document.write( "3<BR>\n" );
document.write( "3.1<BR>\n" );
document.write( "3.4<BR>\n" );
document.write( "3.3<BR>\n" );
document.write( "3.1<BR>\n" );
document.write( "3.4<BR>\n" );
document.write( "4<BR>\n" );
document.write( "3.1<BR>\n" );
document.write( "3.1<BR>\n" );
document.write( "3.1<BR>\n" );
document.write( "2.9<BR>\n" );
document.write( "3.2<BR>\n" );
document.write( "5<BR>\n" );
document.write( "3.2<BR>\n" );
document.write( "3.2<BR>\n" );
document.write( "2.8<BR>\n" );
document.write( "2.7<BR>\n" );
document.write( "3.5<BR>\n" );
document.write( "6<BR>\n" );
document.write( "3.3<BR>\n" );
document.write( "3.4<BR>\n" );
document.write( "3.1<BR>\n" );
document.write( "2.9<BR>\n" );
document.write( "3.4<BR>\n" );
document.write( "7<BR>\n" );
document.write( "3.1<BR>\n" );
document.write( "3.5<BR>\n" );
document.write( "3.2<BR>\n" );
document.write( "3.0<BR>\n" );
document.write( "3.6<BR>\n" );
document.write( " <BR>\n" );
document.write( "​Use the data to construct the chart and R-chart​​<BR>\n" );
document.write( "​<BR>\n" );
document.write( "2. Two machines P and Q are used to produce bags of cement of masses in kilogrammes shown in the table.<BR>\n" );
document.write( " <BR>\n" );
document.write( "Machine P<BR>\n" );
document.write( "50<BR>\n" );
document.write( "51<BR>\n" );
document.write( "48<BR>\n" );
document.write( "50<BR>\n" );
document.write( "51<BR>\n" );
document.write( "52<BR>\n" );
document.write( "53<BR>\n" );
document.write( "51<BR>\n" );
document.write( "51<BR>\n" );
document.write( "Machine Q<BR>\n" );
document.write( "54<BR>\n" );
document.write( "49<BR>\n" );
document.write( "56<BR>\n" );
document.write( "47<BR>\n" );
document.write( "50<BR>\n" );
document.write( "51<BR>\n" );
document.write( "52<BR>\n" );
document.write( "53<BR>\n" );
document.write( " <BR>\n" );
document.write( "Test if there is a difference between the two machines.​<BR>\n" );
document.write( "​C.A.T 2<BR>\n" );
document.write( "1. The probability that a life bulb will have a life time of more than 682 hours is 0.9788. The probability that a bulb will have a life time of more than 703 hours is 0.0051. Find the probability that a bulb will last for more than 648 hours.    <BR>\n" );
document.write( "2. The mean and variance of defective items is 0.72 and 0.6876. Find the probability of getting 12 non-defective items.​​​​        <BR>\n" );
document.write( " <BR>\n" );
document.write( "3. A random sample of householders were classified using the characteristics namely Digital TV and Smart TV and results recorded as follows:<BR>\n" );
document.write( " <BR>\n" );
document.write( " <BR>\n" );
document.write( "Owner occupier<BR>\n" );
document.write( "Council Tenant<BR>\n" );
document.write( "Private Tenant<BR>\n" );
document.write( "Smart TV<BR>\n" );
document.write( "94<BR>\n" );
document.write( "67<BR>\n" );
document.write( "53<BR>\n" );
document.write( "Digital TV<BR>\n" );
document.write( "130<BR>\n" );
document.write( "111<BR>\n" );
document.write( "45<BR>\n" );
document.write( "Black/ White TV<BR>\n" );
document.write( "80<BR>\n" );
document.write( "65<BR>\n" );
document.write( "26<BR>\n" );
document.write( " <BR>\n" );
document.write( "Test for independent at 5%. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848958 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1987)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=848958\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Certainly, let's break down the assignments and guide you through the solutions.\r<BR>\n" );
document.write( "\n" );
document.write( "**Assignment 3**\r<BR>\n" );
document.write( "\n" );
document.write( "**1. Constructing X-bar and R-charts**\r<BR>\n" );
document.write( "\n" );
document.write( "* **Calculate the mean (X̄) and range (R) for each group:**<BR>\n" );
document.write( "    * For each group, calculate the mean of the 5 thickness measurements.<BR>\n" );
document.write( "    * For each group, calculate the range (difference between the highest and lowest values).\r<BR>\n" );
document.write( "\n" );
document.write( "* **Calculate the average of means (X̄̄) and average of ranges (R̄):**<BR>\n" );
document.write( "    * Find the average of the means calculated for each group.<BR>\n" );
document.write( "    * Find the average of the ranges calculated for each group.\r<BR>\n" );
document.write( "\n" );
document.write( "* **Determine control limits:**<BR>\n" );
document.write( "    * **X-bar chart:**<BR>\n" );
document.write( "        * Upper Control Limit (UCL) = X̄̄ + A2 * R̄ <BR>\n" );
document.write( "        * Lower Control Limit (LCL) = X̄̄ - A2 * R̄<BR>\n" );
document.write( "    * **R-chart:**<BR>\n" );
document.write( "        * Upper Control Limit (UCL) = D4 * R̄<BR>\n" );
document.write( "        * Lower Control Limit (LCL) = D3 * R̄\r<BR>\n" );
document.write( "\n" );
document.write( "    * A2, D3, and D4 are constants obtained from control chart constants tables based on the sample size (n = 5 in this case).\r<BR>\n" );
document.write( "\n" );
document.write( "* **Plot the data:**<BR>\n" );
document.write( "    * Plot the group means on the X-bar chart.<BR>\n" );
document.write( "    * Plot the group ranges on the R-chart.\r<BR>\n" );
document.write( "\n" );
document.write( "* **Analyze the charts:**<BR>\n" );
document.write( "    * Check if any points fall outside the control limits. <BR>\n" );
document.write( "    * Look for any patterns or trends in the data.\r<BR>\n" );
document.write( "\n" );
document.write( "**2. Testing for a Difference Between Machines P and Q**\r<BR>\n" );
document.write( "\n" );
document.write( "* **Calculate the means and standard deviations for each machine.**<BR>\n" );
document.write( "* **Perform a t-test:**<BR>\n" );
document.write( "    * **Hypothesis:**<BR>\n" );
document.write( "        * Null hypothesis (H0): The means of the two machines are equal.<BR>\n" );
document.write( "        * Alternative hypothesis (H1): The means of the two machines are not equal.<BR>\n" );
document.write( "    * **Calculate the t-statistic:** <BR>\n" );
document.write( "        * Use the formula for the two-sample t-test, taking into account the possibility of unequal variances.<BR>\n" );
document.write( "    * **Determine the degrees of freedom.**<BR>\n" );
document.write( "    * **Find the critical t-value** based on the degrees of freedom and the chosen significance level (e.g., 0.05 for a 95% confidence level).<BR>\n" );
document.write( "    * **Compare the calculated t-statistic to the critical t-value.**<BR>\n" );
document.write( "    * **Make a decision:** <BR>\n" );
document.write( "        * If the calculated t-statistic is greater than the critical t-value or less than the negative of the critical t-value, reject the null hypothesis. <BR>\n" );
document.write( "        * Otherwise, fail to reject the null hypothesis.\r<BR>\n" );
document.write( "\n" );
document.write( "**C.A.T 2**\r<BR>\n" );
document.write( "\n" );
document.write( "**1. Finding the Probability of a Bulb Lasting More Than 648 Hours**\r<BR>\n" );
document.write( "\n" );
document.write( "* **Use the given probabilities:**<BR>\n" );
document.write( "    * P(lifetime > 682 hours) = 0.9788<BR>\n" );
document.write( "    * P(lifetime > 703 hours) = 0.0051\r<BR>\n" );
document.write( "\n" );
document.write( "* **Determine the probability of a bulb lasting between 682 and 703 hours:**<BR>\n" );
document.write( "    * P(682 < lifetime <= 703) = P(lifetime > 682 hours) - P(lifetime > 703 hours) \r<BR>\n" );
document.write( "\n" );
document.write( "* **Assume a continuous distribution (e.g., exponential or Weibull - this may need to be specified in the problem context).**<BR>\n" );
document.write( "* **Use the cumulative distribution function (CDF) of the assumed distribution** to find the probability of a bulb lasting more than 648 hours. \r<BR>\n" );
document.write( "\n" );
document.write( "**2. Finding the Probability of Getting 12 Non-Defective Items**\r<BR>\n" );
document.write( "\n" );
document.write( "* **Determine the probability of a single item being non-defective:**<BR>\n" );
document.write( "    * If the mean of defective items is 0.72, the mean of non-defective items is 1 - 0.72 = 0.28.\r<BR>\n" );
document.write( "\n" );
document.write( "* **Use the Poisson distribution:** <BR>\n" );
document.write( "    * The Poisson distribution models the probability of a given number of events occurring within a fixed interval of time or space, given the average rate of occurrence.\r<BR>\n" );
document.write( "\n" );
document.write( "    * **Calculate the probability of getting 12 non-defective items using the Poisson probability mass function.**\r<BR>\n" );
document.write( "\n" );
document.write( "**3. Testing for Independence in the TV Ownership Data**\r<BR>\n" );
document.write( "\n" );
document.write( "* **Perform a Chi-Square test of independence:**\r<BR>\n" );
document.write( "\n" );
document.write( "    * **Create a contingency table** summarizing the data.<BR>\n" );
document.write( "    * **Calculate the expected frequencies** for each cell in the table under the assumption of independence.<BR>\n" );
document.write( "    * **Calculate the Chi-Square statistic:** <BR>\n" );
document.write( "        * Sum the squared differences between the observed and expected frequencies, divided by the expected frequencies.<BR>\n" );
document.write( "    * **Determine the degrees of freedom:**<BR>\n" );
document.write( "        * (Number of rows - 1) * (Number of columns - 1)<BR>\n" );
document.write( "    * **Find the critical Chi-Square value** based on the degrees of freedom and the chosen significance level (5% in this case).<BR>\n" );
document.write( "    * **Compare the calculated Chi-Square statistic to the critical Chi-Square value.**<BR>\n" );
document.write( "    * **Make a decision:** <BR>\n" );
document.write( "        * If the calculated Chi-Square statistic is greater than the critical Chi-Square value, reject the null hypothesis of independence. <BR>\n" );
document.write( "        * Otherwise, fail to reject the null hypothesis.\r<BR>\n" );
document.write( "\n" );
document.write( "**Note:**\r<BR>\n" );
document.write( "\n" );
document.write( "* This is a general outline. You'll need to use statistical software (like R, Python, or Excel) or statistical tables to perform the calculations and make the final decisions.<BR>\n" );
document.write( "* Ensure you understand the underlying statistical concepts and assumptions before proceeding with the calculations.\r<BR>\n" );
document.write( "\n" );
document.write( "I hope this helps! Let me know if you have any further questions or need more specific guidance on any of the parts.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );