{"id":2380,"date":"2024-06-29T15:11:37","date_gmt":"2024-06-29T19:11:37","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/?post_type=chapter&#038;p=2380"},"modified":"2024-07-01T02:31:33","modified_gmt":"2024-07-01T06:31:33","slug":"example-of-quality-control-for-space-fasteners","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/chapter\/example-of-quality-control-for-space-fasteners\/","title":{"raw":"Applied Example where \u03c3 is Unknown","rendered":"Applied Example where \u03c3 is Unknown"},"content":{"raw":"<div class=\"textbox textbox--learning-objectives\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Learning Objectives<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nIn this section, we will do the following calculations to estimate the true mean when the population standard deviation (\u03c3) is unknown:\r\n<ul>\r\n \t<li>Review an applied example where confidence intervals are well suited<\/li>\r\n \t<li>Compare to the previous example where confidence intervals were not well suited<\/li>\r\n \t<li>Calculate the lower and upper limits of the confidence interval<\/li>\r\n \t<li>Calculate the required sample size given a required maximum margin of error<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\nIn the previous section, we purposefully used an example that did poorly at predicting future demand. In contrast, in this section, we will choose an example that is well suited to using confidence intervals to analyze the quality control on the product(s) produced.\r\n<h2>Quality Control on Space Fasteners<\/h2>\r\nWe will look at quality control for <a href=\"https:\/\/fastenerandfixing.com\/application-technology\/space-fasteners-play-their-part\/\">space fasteners<\/a> that could be supplied to organizations and companies like <a href=\"https:\/\/standards.nasa.gov\/standard\/NASA\/NASA-STD-6008\">NASA<\/a>, <a href=\"https:\/\/www.spacex.com\/\">SpaceX<\/a> and other companies that build satellites and other machines used in space. These companies must have very strict quality control and adherence to requirements policies.\r\n<table class=\"no-lines\" style=\"border-collapse: collapse;width: 100%\" border=\"0\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 50%\">\r\n\r\n[caption id=\"attachment_2345\" align=\"alignnone\" width=\"600\"]<a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Space_fasteners.jpg\"><img class=\"wp-image-2345 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Space_fasteners.jpg\" alt=\"Image of nuts, bolts and other fasteners that could be used in space.\" width=\"600\" height=\"397\" \/><\/a> Figure 51.1 Possible space fasteners[\/caption]<\/td>\r\n<td style=\"width: 50%\">\r\n\r\n[caption id=\"attachment_2346\" align=\"alignnone\" width=\"600\"]<a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Satellite_image.jpg\"><img class=\"wp-image-2346 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Satellite_image.jpg\" alt=\"Image of satellite orbiting over Earth.\" width=\"600\" height=\"397\" \/><\/a> Figure 51.2 Satellite orbiting over Earth[\/caption]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nSpace fasteners, which would be machine-produced should be fairly uniform and doing analysis based off of their sample statistics such as mean and standard deviation is very appropriate. The parts should have little variation in dimensions and standard deviations would be a great measure to determine the level of variation in the the parts' dimensions, materials' performance and so on.\r\n<h1>Constructing a Confidence Interval using T.INV.2T (VIDEO)<\/h1>\r\nWe will produce confidence intervals for several types of space fasteners in the examples below. In this first example, we will use Excel's <a href=\"https:\/\/support.microsoft.com\/en-us\/office\/t-inv-2t-function-ce72ea19-ec6c-4be7-bed2-b9baf2264f17\">T.INV.2T()<\/a> function.\r\n<h2>Example 51.1<\/h2>\r\n<span style=\"color: #003366\"><strong>Problem Setup:<\/strong><\/span> A company supplies Standard Hexagon Head Cap Screws. They perform regular quality control checks by sampling some of the screws produced and measuring and testing the screws:\r\n\r\n[caption id=\"attachment_2411\" align=\"aligncenter\" width=\"333\"]<a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/HexBoltScrew.jpg\"><img class=\" wp-image-2411\" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/HexBoltScrew.jpg\" alt=\"Picture with two hex bolt screws\" width=\"333\" height=\"286\" \/><\/a> Figure 51.3 Military standard hexagon head cap screws[\/caption]\r\n<ul>\r\n \t<li>They sample 100 screws.<\/li>\r\n \t<li>The mean length of the screws sampled is 0.6245 inches.<\/li>\r\n \t<li>The standard deviation of the lengths is 0.0124 inches.<\/li>\r\n<\/ul>\r\n<span style=\"color: #003366\"><strong>Question<\/strong><\/span>: Construct the 99% confidence interval for the screw lengths. What are the highest and lowest average screw lengths with 99% certainty?\r\n\r\n<span style=\"color: #003366\"><strong>Solution<\/strong><\/span>: <a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example51-1_T-INV-2T_Solutions-1.xlsx\">Click here<\/a>\u00a0to download the Excel file shown in the video below.\r\n\r\nhttps:\/\/youtu.be\/iNqmjYu9lsw\r\n\r\n<strong>Conclusion<\/strong>: There is a 99% chance that the average bolt length is between 0.6212 and 0.6277 inches in length.\r\n<div>\r\n<h1>Click here to reveal the written solutions from the above video<\/h1>\r\n<h2>Solving for the T-Score<\/h2>\r\nLet us first write out the values needed for T.INV.2T:\r\n<ul>\r\n \t<li>[latex]n = 100 [\/latex]<\/li>\r\n \t<li>[latex]df = n - 1 = 100 - 1 = 99 [\/latex]<\/li>\r\n \t<li>[latex]\\alpha = 1-0.99 = 0.01 [\/latex]<\/li>\r\n<\/ul>\r\nThis gives:\u00a0[latex]t = \\text{T.INV.2T}(\\alpha, df) = \\text{T.INV.2T}(0.01, 99) = 2.6264 [\/latex]\r\n<h2>Building up the Confidence Interval<\/h2>\r\nTo build the confidence interval, we use the [latex]t[\/latex]-score above and the following values:\r\n<ul>\r\n \t<li>[latex]\\bar{x} = 0.6245 [\/latex]<\/li>\r\n \t<li>[latex]s = 0.0124 [\/latex]<\/li>\r\n<\/ul>\r\nThis gives:\r\n<ul>\r\n \t<li><span style=\"font-size: 20.5333px;text-align: initial\">[latex] CL_{Lower} = \\bar{x}-t \\cdot \\frac{s}{\\sqrt{n}} = 0.6245 - 2.6264 \\cdot \\frac{0.0124}{\\sqrt{100}} = 0.6245 - 0.0033 = 0.6212 [\/latex]<\/span><\/li>\r\n \t<li><span style=\"font-size: 20.5333px;text-align: initial\">[latex] CL_{Upper} = \\bar{x}+t \\cdot \\frac{s}{\\sqrt{n}} = 0.6245 + 2.6264 \\cdot \\frac{0.0124}{\\sqrt{100}} = 0.6245 + 0.0033 = 0.6277 [\/latex]<\/span><\/li>\r\n<\/ul>\r\n<h2>Conclusion<\/h2>\r\nThere is a 99% chance that the average screw length is between 0.6212 and 0.6277 inches in length.\r\n\r\n<\/div>\r\n<h1>Constructing a Confidence Interval using T.INV (Video)<\/h1>\r\nLet us next look at an example where quality control is performed on shear bolts. In this example, we will use Excel's <a href=\"https:\/\/support.microsoft.com\/en-us\/office\/t-inv-function-2908272b-4e61-4942-9df9-a25fec9b0e2e\">T.INV()<\/a> to calculate the [latex]t[\/latex]-score.\r\n<h2>Example 51.2<\/h2>\r\n<span style=\"color: #003366\"><strong>Problem Setup<\/strong><\/span>: An large order of shear bolts has been received at a spacecraft manufacturing company.\r\n\r\n[caption id=\"attachment_2425\" align=\"aligncenter\" width=\"334\"]<a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/ShearBolts-e1719727581450.jpg\"><img class=\"wp-image-2425 \" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/ShearBolts-e1719727581450.jpg\" alt=\"Image with two shear bolts shown.\" width=\"334\" height=\"263\" \/><\/a> Figure 51.4 NAS-6703 Shear bolts[\/caption]\r\n\r\nThey perform quality control on the order received before paying the manufacturer for the order. They find the following sample statistics based off of the 400 shear bolts they sampled:\r\n<ul>\r\n \t<li>The average bolt length is 0.6251 inches<\/li>\r\n \t<li>The standard deviation of the bolt lengths is 0.0019 inches.<\/li>\r\n<\/ul>\r\nThe quality control team constructs the 90% confidence interval for the bolt lengths based off of the sample statistics they found.\r\n\r\n<span style=\"color: #003366\"><strong>Question<\/strong><\/span>: What confidence interval limits would the team have calculated?\r\n\r\n<span style=\"color: #003366\"><strong>Solution<\/strong><\/span>: <a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example51-2_T-INV_Solutions.xlsx\">Click here<\/a> to download the Excel file shown in the video below.\r\n\r\nhttps:\/\/youtu.be\/-GYTQ0hOErw\r\n\r\n<span style=\"color: #003366\"><strong>Conclusion<\/strong><\/span>: The true average length of the hexaton head cap screws is between 0.6212 and 0.6277 inches with 99% certainty.\r\n<div>\r\n<h1>Click here to reveal the solutions shown in the video above.<\/h1>\r\n<h2>Solving for the T-Score<\/h2>\r\nLet us first write out the values needed for T.INV:\r\n<ul>\r\n \t<li>[latex]n = 400 [\/latex]<\/li>\r\n \t<li>[latex]df = n - 1 = 400 - 1 = 399 [\/latex]<\/li>\r\n \t<li>[latex]\\alpha = 1-0.90 = 0.10 [\/latex]<\/li>\r\n \t<li>[latex]\\frac{\\alpha}{2} = \\frac{0.10}{2} = 0.05 [\/latex]<\/li>\r\n<\/ul>\r\nThis gives: [latex]t = \\text{T.INV}(\\frac{\\alpha}{2}, df) = \\text{T.INV}(0.05, 399) = 1.6487 [\/latex]\r\n<h2>Building up the Confidence Interval<\/h2>\r\nTo build the confidence interval, we use the [latex]t[\/latex]-score above and the following values:\r\n<ul>\r\n \t<li>[latex]\\bar{x} = 0.6251 [\/latex]<\/li>\r\n \t<li>[latex]s = 0.0019 [\/latex]<\/li>\r\n<\/ul>\r\nThis gives:\r\n<ul>\r\n \t<li><span style=\"font-size: 20.5333px;text-align: initial\">[latex] CL_{Lower} = \\bar{x}-t \\cdot \\frac{s}{\\sqrt{n}} = 0.6251 - 1.6487 \\cdot \\frac{0.0019}{\\sqrt{400}} = 0.6251 - 0.00015 = 0.62495 [\/latex]<\/span><\/li>\r\n \t<li><span style=\"font-size: 20.5333px;text-align: initial\">[latex] CL_{Upper} = \\bar{x}+t \\cdot \\frac{s}{\\sqrt{n}} = 0.6251 + 1.6487 \\cdot \\frac{0.0019}{\\sqrt{400}} = 0.6251 + 0.00015 = 0.62525 [\/latex]<\/span><\/li>\r\n<\/ul>\r\n<h2>Conclusion<\/h2>\r\nThere is a 90% chance that the average bolt length is between 0.62495 and 0.62525 inches in length.\r\n\r\n<\/div>\r\n<h1>Constructing a Confidence Interval using CONFIDENCE.T (Video)<\/h1>\r\nLet us revisit the previous example but now use Excel's CONFIDENCE.T() function to solve for the margin of error all in one step (instead of solving for the [latex]t[\/latex]-score and then calculating the margin of error).\r\n<h2>Example 51.3<\/h2>\r\n<span style=\"color: #003366\"><strong>Problem Setup<\/strong><\/span>: Again, we are doing quality control on our large sample of shear bolts. The following is true:\r\n<ul>\r\n \t<li>[latex]\\bar{x} = 0.6251 [\/latex]<\/li>\r\n \t<li>[latex]s = 0.00119 [\/latex]<\/li>\r\n \t<li>[latex]n = 400 [\/latex]<\/li>\r\n \t<li>[latex]\\text{Confidence Level} = 90\\% [\/latex]<\/li>\r\n \t<li>[latex]\\alpha = 10\\% [\/latex]<\/li>\r\n<\/ul>\r\n<span style=\"color: #003366\"><strong>Question:<\/strong><\/span> Can you use CONFIDENCE.T instead, to calculate the lower and upper limits for the 90% confidence interval?\r\n\r\n<span style=\"color: #003366\"><strong>Solution:<\/strong><\/span> <a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example51-3_CONFIDENCE-T_Solutions.xlsx\">Click here<\/a> to download the Excel file shown in the video below.\r\n\r\nhttps:\/\/youtu.be\/XySZryPymog\r\n<div>\r\n<h1>Click here to reveal the written solutions from the above video<\/h1>\r\n<h2>Solving for the Margin of Error (E)<\/h2>\r\nLet us first write out the values needed for CONFIDENCE.T:\r\n<ul>\r\n \t<li>[latex]\\alpha = 1-0.90 = 0.10 [\/latex]<\/li>\r\n \t<li>[latex]s = 0.0019 [\/latex]<\/li>\r\n \t<li>[latex]n = 400 [\/latex]<\/li>\r\n<\/ul>\r\nThis gives: [latex]E = \\text{CONFIDENCE.T}(\\alpha, s, n) = \\text{CONFIDENCE.T}(0.10, 0.0019, 400) = 0.00015 [\/latex]\r\n<h2>Building up the Confidence Interval<\/h2>\r\nTo build the confidence interval, we add and subtract the Margin of Error (E) from the sample mean (x\u0304):\r\n\r\n[latex] CL_{Lower} = \\bar{x}-E = 0.6251 - 0.00015 = 0.62495 [\/latex]\r\n[latex] CL_{Upper} = \\bar{x}+E = 0.6251 + 0.00015 = 0.62525 [\/latex]\r\n<h2>Conclusion<\/h2>\r\nThere is a 90% chance that the average bolt length is between 0.62495 and 0.62525 inches in length.\r\n\r\n<\/div>\r\n<h1>Calculating the Required Sample Size (Video)<\/h1>\r\nLet us revisit the first example in this section where we were performing quality control on the <a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/HexBoltScrew.jpg\">hexagon head cap screws<\/a>. In the case where we need to improve the accuracy of the sample, we will need to increase the sample size. In the next example, we will calculate this new sample size required. The formula is given below:\r\n\r\n\\[n = \\left( \\frac{z \\cdot s}{E} \\right) ^2 \\]\r\n\r\nNote: We will need to use a [latex]z[\/latex]-score instead of a [latex]t[\/latex]-score in this calculation. This is partly because we would need a sample size to calculate a [latex]t[\/latex]-score but the sample size is exactly what we do not have and need to calculate!\r\n<h2>Example 51.4<\/h2>\r\n<span style=\"color: #003366\"><strong>Problem Setup<\/strong><\/span>: The quality control team has stated that the margin of error is too large for your sample of hexagon head cap screws. They would like to reduce this margin of error to 0.002 inches at the 99% confidence level.\r\n\r\n<span style=\"color: #003366\"><strong>Question<\/strong><\/span>: How many more screws do they nee to sample in order to achieve this goal?\r\n\r\n<span style=\"color: #003366\"><strong>Solution<\/strong><\/span>: <a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example51-4_Sample-Size_Solutions.xlsx\">Click here<\/a> to download the Excel file shown in the video below.\r\n\r\nhttps:\/\/youtu.be\/hcv1mdVppy8\r\n\r\n<span style=\"color: #003366\"><strong>Conclusion<\/strong><\/span>: We will need to sample an additional 157 screws in order to reduce the margin of error below 0.002 inches.\r\n<div>\r\n<h1>Click here to reveal the written solutions from the above video<\/h1>\r\n<h2>Finding the z-score<\/h2>\r\nWe first need to find the z-score required for this calculation:\r\n<ul>\r\n \t<li>[latex]\\text{Confidence Level} = 99\\% [\/latex]<\/li>\r\n \t<li>[latex]\\alpha = 1-0.99 = 0.01 = 1\\% [\/latex]<\/li>\r\n \t<li>[latex]\\frac{\\alpha}{2} = \\frac{0.01}{2} = 0.005 [\/latex]<\/li>\r\n \t<li>[latex]z = \\text{NORM.S.INV}(\\frac{\\alpha}{2}) = -2.5758 [\/latex]<\/li>\r\n<\/ul>\r\nWe can drop the minus sign and plug the [latex]z[\/latex]-score into the sample size formula (although it doesn't make any difference when it is squared in the sample size calculation):\r\n\r\n\\[n = \\left( \\frac{z \\cdot s}{E} \\right) ^2\u00a0 = \\left(\\frac{2.5758 \\cdot 0.0124}{0.002} \\right)^2 = 256.58 \\]\r\n\r\nWe always round [latex]n[\/latex] up to the next whole number. This gives the additional number of screws required to be sampled at:\r\n\r\n[latex]\\text{Additional number sampled} = 257 - 100 = 157 [\/latex]\r\n<h2>Conclusion<\/h2>\r\nAn additional 157 screws need to be sampled in order to reduce the margin of error to 0.002 at the 99% confidence level.\r\n\r\n<\/div>","rendered":"<div class=\"textbox textbox--learning-objectives\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Learning Objectives<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>In this section, we will do the following calculations to estimate the true mean when the population standard deviation (\u03c3) is unknown:<\/p>\n<ul>\n<li>Review an applied example where confidence intervals are well suited<\/li>\n<li>Compare to the previous example where confidence intervals were not well suited<\/li>\n<li>Calculate the lower and upper limits of the confidence interval<\/li>\n<li>Calculate the required sample size given a required maximum margin of error<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<p>In the previous section, we purposefully used an example that did poorly at predicting future demand. In contrast, in this section, we will choose an example that is well suited to using confidence intervals to analyze the quality control on the product(s) produced.<\/p>\n<h2>Quality Control on Space Fasteners<\/h2>\n<p>We will look at quality control for <a href=\"https:\/\/fastenerandfixing.com\/application-technology\/space-fasteners-play-their-part\/\">space fasteners<\/a> that could be supplied to organizations and companies like <a href=\"https:\/\/standards.nasa.gov\/standard\/NASA\/NASA-STD-6008\">NASA<\/a>, <a href=\"https:\/\/www.spacex.com\/\">SpaceX<\/a> and other companies that build satellites and other machines used in space. These companies must have very strict quality control and adherence to requirements policies.<\/p>\n<table class=\"no-lines\" style=\"border-collapse: collapse;width: 100%\">\n<tbody>\n<tr>\n<td style=\"width: 50%\">\n<figure id=\"attachment_2345\" aria-describedby=\"caption-attachment-2345\" style=\"width: 600px\" class=\"wp-caption alignnone\"><a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Space_fasteners.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2345 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Space_fasteners.jpg\" alt=\"Image of nuts, bolts and other fasteners that could be used in space.\" width=\"600\" height=\"397\" srcset=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Space_fasteners.jpg 600w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Space_fasteners-300x199.jpg 300w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Space_fasteners-65x43.jpg 65w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Space_fasteners-225x149.jpg 225w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Space_fasteners-350x232.jpg 350w\" sizes=\"auto, (max-width: 600px) 100vw, 600px\" \/><\/a><figcaption id=\"caption-attachment-2345\" class=\"wp-caption-text\">Figure 51.1 Possible space fasteners<\/figcaption><\/figure>\n<\/td>\n<td style=\"width: 50%\">\n<figure id=\"attachment_2346\" aria-describedby=\"caption-attachment-2346\" style=\"width: 600px\" class=\"wp-caption alignnone\"><a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Satellite_image.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2346 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Satellite_image.jpg\" alt=\"Image of satellite orbiting over Earth.\" width=\"600\" height=\"397\" srcset=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Satellite_image.jpg 600w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Satellite_image-300x199.jpg 300w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Satellite_image-65x43.jpg 65w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Satellite_image-225x149.jpg 225w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Satellite_image-350x232.jpg 350w\" sizes=\"auto, (max-width: 600px) 100vw, 600px\" \/><\/a><figcaption id=\"caption-attachment-2346\" class=\"wp-caption-text\">Figure 51.2 Satellite orbiting over Earth<\/figcaption><\/figure>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Space fasteners, which would be machine-produced should be fairly uniform and doing analysis based off of their sample statistics such as mean and standard deviation is very appropriate. The parts should have little variation in dimensions and standard deviations would be a great measure to determine the level of variation in the the parts&#8217; dimensions, materials&#8217; performance and so on.<\/p>\n<h1>Constructing a Confidence Interval using T.INV.2T (VIDEO)<\/h1>\n<p>We will produce confidence intervals for several types of space fasteners in the examples below. In this first example, we will use Excel&#8217;s <a href=\"https:\/\/support.microsoft.com\/en-us\/office\/t-inv-2t-function-ce72ea19-ec6c-4be7-bed2-b9baf2264f17\">T.INV.2T()<\/a> function.<\/p>\n<h2>Example 51.1<\/h2>\n<p><span style=\"color: #003366\"><strong>Problem Setup:<\/strong><\/span> A company supplies Standard Hexagon Head Cap Screws. They perform regular quality control checks by sampling some of the screws produced and measuring and testing the screws:<\/p>\n<figure id=\"attachment_2411\" aria-describedby=\"caption-attachment-2411\" style=\"width: 333px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/HexBoltScrew.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2411\" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/HexBoltScrew.jpg\" alt=\"Picture with two hex bolt screws\" width=\"333\" height=\"286\" srcset=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/HexBoltScrew.jpg 330w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/HexBoltScrew-300x257.jpg 300w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/HexBoltScrew-65x56.jpg 65w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/HexBoltScrew-225x193.jpg 225w\" sizes=\"auto, (max-width: 333px) 100vw, 333px\" \/><\/a><figcaption id=\"caption-attachment-2411\" class=\"wp-caption-text\">Figure 51.3 Military standard hexagon head cap screws<\/figcaption><\/figure>\n<ul>\n<li>They sample 100 screws.<\/li>\n<li>The mean length of the screws sampled is 0.6245 inches.<\/li>\n<li>The standard deviation of the lengths is 0.0124 inches.<\/li>\n<\/ul>\n<p><span style=\"color: #003366\"><strong>Question<\/strong><\/span>: Construct the 99% confidence interval for the screw lengths. What are the highest and lowest average screw lengths with 99% certainty?<\/p>\n<p><span style=\"color: #003366\"><strong>Solution<\/strong><\/span>: <a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example51-1_T-INV-2T_Solutions-1.xlsx\">Click here<\/a>\u00a0to download the Excel file shown in the video below.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"How to build a confidence interval using T.INV.2T (for the true mean where sigma is unknown)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/iNqmjYu9lsw?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p><strong>Conclusion<\/strong>: There is a 99% chance that the average bolt length is between 0.6212 and 0.6277 inches in length.<\/p>\n<div>\n<h1>Click here to reveal the written solutions from the above video<\/h1>\n<h2>Solving for the T-Score<\/h2>\n<p>Let us first write out the values needed for T.INV.2T:<\/p>\n<ul>\n<li>[latex]n = 100[\/latex]<\/li>\n<li>[latex]df = n - 1 = 100 - 1 = 99[\/latex]<\/li>\n<li>[latex]\\alpha = 1-0.99 = 0.01[\/latex]<\/li>\n<\/ul>\n<p>This gives:\u00a0[latex]t = \\text{T.INV.2T}(\\alpha, df) = \\text{T.INV.2T}(0.01, 99) = 2.6264[\/latex]<\/p>\n<h2>Building up the Confidence Interval<\/h2>\n<p>To build the confidence interval, we use the [latex]t[\/latex]-score above and the following values:<\/p>\n<ul>\n<li>[latex]\\bar{x} = 0.6245[\/latex]<\/li>\n<li>[latex]s = 0.0124[\/latex]<\/li>\n<\/ul>\n<p>This gives:<\/p>\n<ul>\n<li><span style=\"font-size: 20.5333px;text-align: initial\">[latex]CL_{Lower} = \\bar{x}-t \\cdot \\frac{s}{\\sqrt{n}} = 0.6245 - 2.6264 \\cdot \\frac{0.0124}{\\sqrt{100}} = 0.6245 - 0.0033 = 0.6212[\/latex]<\/span><\/li>\n<li><span style=\"font-size: 20.5333px;text-align: initial\">[latex]CL_{Upper} = \\bar{x}+t \\cdot \\frac{s}{\\sqrt{n}} = 0.6245 + 2.6264 \\cdot \\frac{0.0124}{\\sqrt{100}} = 0.6245 + 0.0033 = 0.6277[\/latex]<\/span><\/li>\n<\/ul>\n<h2>Conclusion<\/h2>\n<p>There is a 99% chance that the average screw length is between 0.6212 and 0.6277 inches in length.<\/p>\n<\/div>\n<h1>Constructing a Confidence Interval using T.INV (Video)<\/h1>\n<p>Let us next look at an example where quality control is performed on shear bolts. In this example, we will use Excel&#8217;s <a href=\"https:\/\/support.microsoft.com\/en-us\/office\/t-inv-function-2908272b-4e61-4942-9df9-a25fec9b0e2e\">T.INV()<\/a> to calculate the [latex]t[\/latex]-score.<\/p>\n<h2>Example 51.2<\/h2>\n<p><span style=\"color: #003366\"><strong>Problem Setup<\/strong><\/span>: An large order of shear bolts has been received at a spacecraft manufacturing company.<\/p>\n<figure id=\"attachment_2425\" aria-describedby=\"caption-attachment-2425\" style=\"width: 334px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/ShearBolts-e1719727581450.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2425\" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/ShearBolts-e1719727581450.jpg\" alt=\"Image with two shear bolts shown.\" width=\"334\" height=\"263\" srcset=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/ShearBolts-e1719727581450.jpg 520w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/ShearBolts-e1719727581450-300x236.jpg 300w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/ShearBolts-e1719727581450-65x51.jpg 65w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/ShearBolts-e1719727581450-225x177.jpg 225w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/ShearBolts-e1719727581450-350x275.jpg 350w\" sizes=\"auto, (max-width: 334px) 100vw, 334px\" \/><\/a><figcaption id=\"caption-attachment-2425\" class=\"wp-caption-text\">Figure 51.4 NAS-6703 Shear bolts<\/figcaption><\/figure>\n<p>They perform quality control on the order received before paying the manufacturer for the order. They find the following sample statistics based off of the 400 shear bolts they sampled:<\/p>\n<ul>\n<li>The average bolt length is 0.6251 inches<\/li>\n<li>The standard deviation of the bolt lengths is 0.0019 inches.<\/li>\n<\/ul>\n<p>The quality control team constructs the 90% confidence interval for the bolt lengths based off of the sample statistics they found.<\/p>\n<p><span style=\"color: #003366\"><strong>Question<\/strong><\/span>: What confidence interval limits would the team have calculated?<\/p>\n<p><span style=\"color: #003366\"><strong>Solution<\/strong><\/span>: <a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example51-2_T-INV_Solutions.xlsx\">Click here<\/a> to download the Excel file shown in the video below.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Using Excel&#39;s T.INV function to build up a 90% Confidence Interval when Sigma is Unknown\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/-GYTQ0hOErw?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p><span style=\"color: #003366\"><strong>Conclusion<\/strong><\/span>: The true average length of the hexaton head cap screws is between 0.6212 and 0.6277 inches with 99% certainty.<\/p>\n<div>\n<h1>Click here to reveal the solutions shown in the video above.<\/h1>\n<h2>Solving for the T-Score<\/h2>\n<p>Let us first write out the values needed for T.INV:<\/p>\n<ul>\n<li>[latex]n = 400[\/latex]<\/li>\n<li>[latex]df = n - 1 = 400 - 1 = 399[\/latex]<\/li>\n<li>[latex]\\alpha = 1-0.90 = 0.10[\/latex]<\/li>\n<li>[latex]\\frac{\\alpha}{2} = \\frac{0.10}{2} = 0.05[\/latex]<\/li>\n<\/ul>\n<p>This gives: [latex]t = \\text{T.INV}(\\frac{\\alpha}{2}, df) = \\text{T.INV}(0.05, 399) = 1.6487[\/latex]<\/p>\n<h2>Building up the Confidence Interval<\/h2>\n<p>To build the confidence interval, we use the [latex]t[\/latex]-score above and the following values:<\/p>\n<ul>\n<li>[latex]\\bar{x} = 0.6251[\/latex]<\/li>\n<li>[latex]s = 0.0019[\/latex]<\/li>\n<\/ul>\n<p>This gives:<\/p>\n<ul>\n<li><span style=\"font-size: 20.5333px;text-align: initial\">[latex]CL_{Lower} = \\bar{x}-t \\cdot \\frac{s}{\\sqrt{n}} = 0.6251 - 1.6487 \\cdot \\frac{0.0019}{\\sqrt{400}} = 0.6251 - 0.00015 = 0.62495[\/latex]<\/span><\/li>\n<li><span style=\"font-size: 20.5333px;text-align: initial\">[latex]CL_{Upper} = \\bar{x}+t \\cdot \\frac{s}{\\sqrt{n}} = 0.6251 + 1.6487 \\cdot \\frac{0.0019}{\\sqrt{400}} = 0.6251 + 0.00015 = 0.62525[\/latex]<\/span><\/li>\n<\/ul>\n<h2>Conclusion<\/h2>\n<p>There is a 90% chance that the average bolt length is between 0.62495 and 0.62525 inches in length.<\/p>\n<\/div>\n<h1>Constructing a Confidence Interval using CONFIDENCE.T (Video)<\/h1>\n<p>Let us revisit the previous example but now use Excel&#8217;s CONFIDENCE.T() function to solve for the margin of error all in one step (instead of solving for the [latex]t[\/latex]-score and then calculating the margin of error).<\/p>\n<h2>Example 51.3<\/h2>\n<p><span style=\"color: #003366\"><strong>Problem Setup<\/strong><\/span>: Again, we are doing quality control on our large sample of shear bolts. The following is true:<\/p>\n<ul>\n<li>[latex]\\bar{x} = 0.6251[\/latex]<\/li>\n<li>[latex]s = 0.00119[\/latex]<\/li>\n<li>[latex]n = 400[\/latex]<\/li>\n<li>[latex]\\text{Confidence Level} = 90\\%[\/latex]<\/li>\n<li>[latex]\\alpha = 10\\%[\/latex]<\/li>\n<\/ul>\n<p><span style=\"color: #003366\"><strong>Question:<\/strong><\/span> Can you use CONFIDENCE.T instead, to calculate the lower and upper limits for the 90% confidence interval?<\/p>\n<p><span style=\"color: #003366\"><strong>Solution:<\/strong><\/span> <a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example51-3_CONFIDENCE-T_Solutions.xlsx\">Click here<\/a> to download the Excel file shown in the video below.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"How to use CONFIDENCE.T to build up a confidence interval when sigma is unknown\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/XySZryPymog?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div>\n<h1>Click here to reveal the written solutions from the above video<\/h1>\n<h2>Solving for the Margin of Error (E)<\/h2>\n<p>Let us first write out the values needed for CONFIDENCE.T:<\/p>\n<ul>\n<li>[latex]\\alpha = 1-0.90 = 0.10[\/latex]<\/li>\n<li>[latex]s = 0.0019[\/latex]<\/li>\n<li>[latex]n = 400[\/latex]<\/li>\n<\/ul>\n<p>This gives: [latex]E = \\text{CONFIDENCE.T}(\\alpha, s, n) = \\text{CONFIDENCE.T}(0.10, 0.0019, 400) = 0.00015[\/latex]<\/p>\n<h2>Building up the Confidence Interval<\/h2>\n<p>To build the confidence interval, we add and subtract the Margin of Error (E) from the sample mean (x\u0304):<\/p>\n<p>[latex]CL_{Lower} = \\bar{x}-E = 0.6251 - 0.00015 = 0.62495[\/latex]<br \/>\n[latex]CL_{Upper} = \\bar{x}+E = 0.6251 + 0.00015 = 0.62525[\/latex]<\/p>\n<h2>Conclusion<\/h2>\n<p>There is a 90% chance that the average bolt length is between 0.62495 and 0.62525 inches in length.<\/p>\n<\/div>\n<h1>Calculating the Required Sample Size (Video)<\/h1>\n<p>Let us revisit the first example in this section where we were performing quality control on the <a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/HexBoltScrew.jpg\">hexagon head cap screws<\/a>. In the case where we need to improve the accuracy of the sample, we will need to increase the sample size. In the next example, we will calculate this new sample size required. The formula is given below:<\/p>\n<p>\\[n = \\left( \\frac{z \\cdot s}{E} \\right) ^2 \\]<\/p>\n<p>Note: We will need to use a [latex]z[\/latex]-score instead of a [latex]t[\/latex]-score in this calculation. This is partly because we would need a sample size to calculate a [latex]t[\/latex]-score but the sample size is exactly what we do not have and need to calculate!<\/p>\n<h2>Example 51.4<\/h2>\n<p><span style=\"color: #003366\"><strong>Problem Setup<\/strong><\/span>: The quality control team has stated that the margin of error is too large for your sample of hexagon head cap screws. They would like to reduce this margin of error to 0.002 inches at the 99% confidence level.<\/p>\n<p><span style=\"color: #003366\"><strong>Question<\/strong><\/span>: How many more screws do they nee to sample in order to achieve this goal?<\/p>\n<p><span style=\"color: #003366\"><strong>Solution<\/strong><\/span>: <a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/06\/Example51-4_Sample-Size_Solutions.xlsx\">Click here<\/a> to download the Excel file shown in the video below.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-4\" title=\"How to calculate the required sample size to reduce the margin of error for a 99% confidence level.\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/hcv1mdVppy8?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p><span style=\"color: #003366\"><strong>Conclusion<\/strong><\/span>: We will need to sample an additional 157 screws in order to reduce the margin of error below 0.002 inches.<\/p>\n<div>\n<h1>Click here to reveal the written solutions from the above video<\/h1>\n<h2>Finding the z-score<\/h2>\n<p>We first need to find the z-score required for this calculation:<\/p>\n<ul>\n<li>[latex]\\text{Confidence Level} = 99\\%[\/latex]<\/li>\n<li>[latex]\\alpha = 1-0.99 = 0.01 = 1\\%[\/latex]<\/li>\n<li>[latex]\\frac{\\alpha}{2} = \\frac{0.01}{2} = 0.005[\/latex]<\/li>\n<li>[latex]z = \\text{NORM.S.INV}(\\frac{\\alpha}{2}) = -2.5758[\/latex]<\/li>\n<\/ul>\n<p>We can drop the minus sign and plug the [latex]z[\/latex]-score into the sample size formula (although it doesn&#8217;t make any difference when it is squared in the sample size calculation):<\/p>\n<p>\\[n = \\left( \\frac{z \\cdot s}{E} \\right) ^2\u00a0 = \\left(\\frac{2.5758 \\cdot 0.0124}{0.002} \\right)^2 = 256.58 \\]<\/p>\n<p>We always round [latex]n[\/latex] up to the next whole number. This gives the additional number of screws required to be sampled at:<\/p>\n<p>[latex]\\text{Additional number sampled} = 257 - 100 = 157[\/latex]<\/p>\n<h2>Conclusion<\/h2>\n<p>An additional 157 screws need to be sampled in order to reduce the margin of error to 0.002 at the 99% confidence level.<\/p>\n<\/div>\n","protected":false},"author":865,"menu_order":5,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-2380","chapter","type-chapter","status-publish","hentry"],"part":487,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/chapters\/2380","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/wp\/v2\/users\/865"}],"version-history":[{"count":25,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/chapters\/2380\/revisions"}],"predecessor-version":[{"id":2455,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/chapters\/2380\/revisions\/2455"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/parts\/487"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/chapters\/2380\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/wp\/v2\/media?parent=2380"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/chapter-type?post=2380"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/wp\/v2\/contributor?post=2380"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/wp\/v2\/license?post=2380"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}