Question
A product development group determines that it must have a fiber, which among other properties, has a minimum tensile strength of 2.500 gm in 99% of the fiber used.
Manufacturer X offers to supply such fiber and a contract is arranged.
Manufacturer X knows that the standard deviation for the process is 0.020 gm.
What minimum aimed at value of the mean is required to assure compliance with the contract?
Assume the process is in statistical control and the distribution is normal.
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      • Illuminate\Cache\Events\CacheHit (286μs)
      • Illuminate\Cache\Events\CacheHit (183μs)
      • Illuminate\Cache\Events\CacheHit (246μs)
      • Illuminate\Cache\Events\CacheHit (183μs)
      • Illuminate\Cache\Events\CacheHit (233μs)
      • Illuminate\Cache\Events\CacheHit (290μs)
      • Illuminate\Cache\Events\CacheHit (190μs)
      • Illuminate\Cache\Events\CacheHit (326μs)
      • Illuminate\Cache\Events\CacheHit (231μs)
      • Illuminate\Cache\Events\CacheHit (167μs)
      • creating: site.layouts.app (458μs)
      • composing: site.layouts.app (20μs)
      • creating: components.canonical (419μs)
      • composing: components.canonical (156μs)
      • creating: components.open-graph (313μs)
      • composing: components.open-graph (86μs)
      • creating: site.headers.header (321μs)
      • composing: site.headers.header (78μs)
      • Illuminate\Cache\Events\CacheHit (2.3ms)
      • creating: components.footer (168μs)
      • composing: components.footer (97μs)
      • Illuminate\Cache\Events\CacheHit (1.14ms)
      • Illuminate\Cache\Events\CacheMissed (434μs)
      • Illuminate\Database\Events\StatementPrepared (931μs)
      • Illuminate\Database\Events\QueryExecuted (1.1ms)
      • eloquent.retrieved: App\Models\SubjectCat (176μs)
      • Illuminate\Cache\Events\KeyWritten (454μs)
      • Illuminate\Cache\Events\CacheHit (259μs)
      • Illuminate\Cache\Events\CacheHit (139μs)
      • Illuminate\Cache\Events\CacheHit (233μs)
      • Illuminate\Cache\Events\CacheHit (124μs)
      • Illuminate\Cache\Events\CacheHit (126μs)
      • Illuminate\Cache\Events\CacheHit (508μs)
      • Illuminate\Cache\Events\CacheHit (364μs)
      • Illuminate\Cache\Events\CacheHit (211μs)
      • Illuminate\Cache\Events\CacheHit (288μs)
      • Illuminate\Cache\Events\CacheHit (202μs)
      • Illuminate\Cache\Events\CacheHit (221μs)
      • Illuminate\Cache\Events\CacheHit (194μs)
      • creating: components.forms.contact-us (466μs)
      • composing: components.forms.contact-us (199μs)
      • creating: components.forms.get-started (357μs)
      • composing: components.forms.get-started (88μs)
      • creating: components.forms.free-tool-download (211μs)
      • composing: components.forms.free-tool-download (84μs)
      • creating: components.forms.claim-free-worksheet (211μs)
      • composing: components.forms.claim-free-worksheet (197μs)
      • creating: components.forms.tutor-subscription-waitlist (237μs)
      • composing: components.forms.tutor-subscription-waitlist (240μs)
      • creating: components.forms.tutor-subscription-join (161μs)
      • composing: components.forms.tutor-subscription-join (75μs)
      • creating: components.forms.tutor-support (126μs)
      • composing: components.forms.tutor-support (72μs)
      • 311 x Illuminate\Cache\Events\CacheHit (21.05%)
        149ms
        20 x Illuminate\Database\Events\QueryExecuted (5.91%)
        41.80ms
        20 x Illuminate\Database\Events\StatementPrepared (5.43%)
        38.40ms
        11 x Illuminate\Cache\Events\CacheMissed (2.14%)
        15.12ms
        11 x Illuminate\Cache\Events\KeyWritten (1.65%)
        11.64ms
        1 x Illuminate\Foundation\Events\LocaleUpdated (0.73%)
        5.17ms
        1 x eloquent.booting: App\Models\SubjectCat (0.25%)
        1.77ms
        1 x Illuminate\Routing\Events\Routing (0.25%)
        1.74ms
        12 x eloquent.retrieved: App\Models\SubjectCat (0.23%)
        1.65ms
        1 x Illuminate\Database\Events\ConnectionEstablished (0.17%)
        1.22ms
        1 x creating: homework.show (0.15%)
        1.06ms
        2 x eloquent.retrieved: App\Models\Subject (0.14%)
        963μs
        1 x Illuminate\Routing\Events\RouteMatched (0.09%)
        632μs
        1 x eloquent.booted: App\Models\SubjectCat (0.08%)
        554μs
        1 x creating: components.breadcrumbs (0.07%)
        481μs
        1 x creating: components.forms.contact-us (0.07%)
        466μs
        1 x creating: site.layouts.app (0.06%)
        458μs
        1 x eloquent.booted: App\Models\HomeworkLibrary\HomeworkLibrary (0.06%)
        429μs
        1 x creating: components.canonical (0.06%)
        419μs
        5 x eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibraryFile (0.06%)
        404μs
        1 x creating: components.forms.get-started (0.05%)
        357μs
        1 x creating: site.headers.header (0.05%)
        321μs
        1 x creating: components.open-graph (0.04%)
        313μs
        1 x composing: components.breadcrumbs (0.04%)
        271μs
        7 x eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibrary (0.04%)
        257μs
        1 x eloquent.booting: App\Models\HomeworkLibrary\HomeworkLibrary (0.04%)
        255μs
        1 x composing: components.forms.tutor-subscription-waitlist (0.03%)
        240μs
        1 x creating: components.forms.tutor-subscription-waitlist (0.03%)
        237μs
        1 x creating: components.forms.free-tool-download (0.03%)
        211μs
        1 x creating: components.forms.claim-free-worksheet (0.03%)
        211μs
        1 x composing: components.forms.contact-us (0.03%)
        199μs
        1 x composing: components.forms.claim-free-worksheet (0.03%)
        197μs
        1 x composing: homework.show (0.03%)
        184μs
        1 x eloquent.booting: App\Models\HomeworkLibrary\HomeworkLibraryFile (0.02%)
        176μs
        1 x creating: components.footer (0.02%)
        168μs
        1 x creating: components.forms.tutor-subscription-join (0.02%)
        161μs
        1 x composing: components.canonical (0.02%)
        156μs
        1 x eloquent.booting: App\Models\Subject (0.02%)
        144μs
        1 x creating: components.forms.tutor-support (0.02%)
        126μs
        1 x composing: components.footer (0.01%)
        97μs
        1 x composing: components.forms.get-started (0.01%)
        88μs
        1 x composing: components.open-graph (0.01%)
        86μs
        1 x composing: components.forms.free-tool-download (0.01%)
        84μs
        1 x composing: site.headers.header (0.01%)
        78μs
        1 x composing: components.forms.tutor-subscription-join (0.01%)
        75μs
        1 x composing: components.forms.tutor-support (0.01%)
        72μs
        1 x eloquent.booted: App\Models\Subject (0.01%)
        63μs
        1 x eloquent.booted: App\Models\HomeworkLibrary\HomeworkLibraryFile (0.01%)
        63μs
        1 x composing: site.layouts.app (0%)
        20μs
      14 templates were rendered
      • 1x homework.showshow.blade.phpblade
      • 1x components.breadcrumbsbreadcrumbs.blade.phpblade
      • 1x site.layouts.appapp.blade.phpblade
      • 1x components.canonicalcanonical.blade.phpblade
      • 1x components.open-graphopen-graph.blade.phpblade
      • 1x site.headers.headerheader.blade.phpblade
      • 1x components.footerfooter.blade.phpblade
      • 1x components.forms.contact-uscontact-us.blade.phpblade
      • 1x components.forms.get-startedget-started.blade.phpblade
      • 1x components.forms.free-tool-downloadfree-tool-download.blade.phpblade
      • 1x components.forms.claim-free-worksheetclaim-free-worksheet.blade.phpblade
      • 1x components.forms.tutor-subscription-waitlisttutor-subscription-waitlist.blade.phpblade
      • 1x components.forms.tutor-subscription-jointutor-subscription-join.blade.phpblade
      • 1x components.forms.tutor-supporttutor-support.blade.phpblade
      uri
      GET college-homework-library/{category}/{subject}/{id}
      middleware
      web, utm.parameters
      controller
      App\Http\Controllers\HomeworkLibraryController@show
      namespace
      where
      as
      homework.show
      file
      app/Http/Controllers/HomeworkLibraryController.php:79-176
      20 statements were executed, 5 of which were duplicates, 15 unique. Show only duplicated44.62ms
      • Connection Establishedtwenty4_siteHomeworkLibraryController.php#91
        Backtrace
        • 13. app/Http/Controllers/HomeworkLibraryController.php:91
        • 14. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 15. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 16. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
        • 17. vendor/laravel/framework/src/Illuminate/Routing/Route.php:205
      • select * from `solutionslibrary` where `status` = 'published' and `price` > 0 and `solutionslibrary`.`id` = '21303' limit 1
        9.05mstwenty4_siteHomeworkLibraryController.php#97
        Bindings
        • 0: published
        • 1: 0
        • 2: 21303
        Backtrace
        • 16. app/Http/Controllers/HomeworkLibraryController.php:97
        • 17. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 18. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 19. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
        • 20. vendor/laravel/framework/src/Illuminate/Routing/Route.php:205
      • select * from `subjects` where `subjects`.`id` in (57)
        1.38mstwenty4_siteHomeworkLibraryController.php#97
        Backtrace
        • 21. app/Http/Controllers/HomeworkLibraryController.php:97
        • 22. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 23. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 24. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
        • 25. vendor/laravel/framework/src/Illuminate/Routing/Route.php:205
      • select * from `solutionslibrary_files` where `solutionslibrary_files`.`solutionlib_id` in (21303)
        2.39mstwenty4_siteHomeworkLibraryController.php#97
        Backtrace
        • 21. app/Http/Controllers/HomeworkLibraryController.php:97
        • 22. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 23. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 24. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
        • 25. vendor/laravel/framework/src/Illuminate/Routing/Route.php:205
      • select * from `subject_cats` where `subject_cats`.`id` = 10 limit 1
        1.13mstwenty4_siteHomeworkLibrary.php#201
        Bindings
        • 0: 10
        Backtrace
        • 20. app/Models/HomeworkLibrary/HomeworkLibrary.php:201
        • 26. app/Http/Controllers/HomeworkLibraryController.php:105
        • 27. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 28. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 29. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
      • select * from `solutionslibrary` where `id` <> 21303 and `subject` = 57 and `status` = 'published' and `price` > 0 order by RAND() limit 6
        11.41mstwenty4_siteHomeworkLibraryRepository.php#30
        Bindings
        • 0: 21303
        • 1: 57
        • 2: published
        • 3: 0
        Backtrace
        • 14. app/Repositories/HomeworkLibraryRepository.php:30
        • 15. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 16. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 18. app/Repositories/HomeworkLibraryRepository.php:39
        • 19. app/Http/Controllers/HomeworkLibraryController.php:139
      • select * from `subjects` where `subjects`.`id` in (57)
        1.07mstwenty4_siteHomeworkLibraryRepository.php#30
        Backtrace
        • 19. app/Repositories/HomeworkLibraryRepository.php:30
        • 20. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 21. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 23. app/Repositories/HomeworkLibraryRepository.php:39
        • 24. app/Http/Controllers/HomeworkLibraryController.php:139
      • select * from `solutionslibrary_files` where `solutionslibrary_files`.`solutionlib_id` = 21303 and `solutionslibrary_files`.`solutionlib_id` is not null and `publish` = 'question' order by `order` asc, `id` asc
        1.12mstwenty4_siteHomeworkLibrary.php#260
        Bindings
        • 0: 21303
        • 1: question
        Backtrace
        • 15. app/Models/HomeworkLibrary/HomeworkLibrary.php:260
        • 16. app/Transformers/HomeworkLibrary/HomeworkLibraryTransformer.php:58
        • 19. vendor/league/fractal/src/TransformerAbstract.php:128
        • 20. vendor/league/fractal/src/TransformerAbstract.php:107
        • 21. vendor/league/fractal/src/Scope.php:383
      • select * from `solutionslibrary_files` where `solutionslibrary_files`.`solutionlib_id` = 21303 and `solutionslibrary_files`.`solutionlib_id` is not null and `publish` = 'teaser' order by `order` asc, `id` asc
        1.01mstwenty4_siteHomeworkLibrary.php#260
        Bindings
        • 0: 21303
        • 1: teaser
        Backtrace
        • 15. app/Models/HomeworkLibrary/HomeworkLibrary.php:260
        • 16. app/Transformers/HomeworkLibrary/HomeworkLibraryTransformer.php:69
        • 19. vendor/league/fractal/src/TransformerAbstract.php:128
        • 20. vendor/league/fractal/src/TransformerAbstract.php:107
        • 21. vendor/league/fractal/src/Scope.php:383
      • select * from `solutionslibrary_files` where `solutionslibrary_files`.`solutionlib_id` = 21303 and `solutionslibrary_files`.`solutionlib_id` is not null and `publish` = 'solution' order by `order` asc, `id` asc
        850μstwenty4_siteHomeworkLibrary.php#260
        Bindings
        • 0: 21303
        • 1: solution
        Backtrace
        • 15. app/Models/HomeworkLibrary/HomeworkLibrary.php:260
        • 16. app/Transformers/HomeworkLibrary/HomeworkLibraryTransformer.php:80
        • 19. vendor/league/fractal/src/TransformerAbstract.php:128
        • 20. vendor/league/fractal/src/TransformerAbstract.php:107
        • 21. vendor/league/fractal/src/Scope.php:383
      • select * from `subject_cats` where `subject_cats`.`id` = 10 limit 1
        1.19mstwenty4_siteHomeworkLibrary.php#201
        Bindings
        • 0: 10
        Backtrace
        • 20. app/Models/HomeworkLibrary/HomeworkLibrary.php:201
        • 32. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
        • 33. vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58
        • 34. vendor/livewire/livewire/src/ComponentConcerns/RendersLivewireComponents.php:69
        • 35. vendor/laravel/framework/src/Illuminate/View/Engines/CompilerEngine.php:70
      • select * from `subject_cats` where `subject_cats`.`id` = 1 limit 1
        1.2mstwenty4_siteSubject.php#100
        Bindings
        • 0: 1
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 3 limit 1
        1.45mstwenty4_siteSubject.php#100
        Bindings
        • 0: 3
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 4 limit 1
        1.43mstwenty4_siteSubject.php#100
        Bindings
        • 0: 4
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 10 limit 1
        3.41mstwenty4_siteSubject.php#100
        Bindings
        • 0: 10
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 33 limit 1
        920μstwenty4_siteSubject.php#100
        Bindings
        • 0: 33
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 11 limit 1
        1.32mstwenty4_siteSubject.php#100
        Bindings
        • 0: 11
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 5 limit 1
        1.12mstwenty4_siteSubject.php#100
        Bindings
        • 0: 5
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 34 limit 1
        1.07mstwenty4_siteSubject.php#100
        Bindings
        • 0: 34
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 31 limit 1
        890μstwenty4_siteSubject.php#100
        Bindings
        • 0: 31
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 36 limit 1
        1.21mstwenty4_siteSubject.php#100
        Bindings
        • 0: 36
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 28. view::components.footer:170
      App\Models\SubjectCat
      12SubjectCat.php
      App\Models\HomeworkLibrary\HomeworkLibrary
      7HomeworkLibrary.php
      App\Models\HomeworkLibrary\HomeworkLibraryFile
      5HomeworkLibraryFile.php
      App\Models\Subject
      2Subject.php
          _token
          UtWZLXa2BNFWuAa9Uw7HVYXzlnhSqByAgKGp09WQ
          utm_source
          direct
          redirectUrl
          /college-homework-library/Engineering/Engineering-Other/21303
          _previous
          array:1 [ "url" => "https://staging.dev.24houranswers.com/college-homework-library/Engineering/Eng...
          _flash
          array:2 [ "old" => [] "new" => [] ]
          PHPDEBUGBAR_STACK_DATA
          []
          path_info
          /college-homework-library/Engineering/Engineering-Other/21303
          status_code
          200
          
          status_text
          OK
          format
          html
          content_type
          text/html; charset=UTF-8
          request_query
          []
          
          request_request
          []
          
          request_headers
          0 of 0
          array:21 [ "priority" => array:1 [ 0 => "u=0, i" ] "accept-encoding" => array:1 [ 0 => "gzip, deflate, br, zstd" ] "sec-fetch-dest" => array:1 [ 0 => "document" ] "sec-fetch-user" => array:1 [ 0 => "?1" ] "sec-fetch-mode" => array:1 [ 0 => "navigate" ] "sec-fetch-site" => array:1 [ 0 => "none" ] "accept" => array:1 [ 0 => "text/html,application/xhtml+xml,application/xml;q=0.9,image/avif,image/webp,image/apng,*/*;q=0.8,application/signed-exchange;v=b3;q=0.7" ] "user-agent" => array:1 [ 0 => "Mozilla/5.0 AppleWebKit/537.36 (KHTML, like Gecko; compatible; ClaudeBot/1.0; +claudebot@anthropic.com)" ] "upgrade-insecure-requests" => array:1 [ 0 => "1" ] "sec-ch-ua-platform" => array:1 [ 0 => ""Windows"" ] "sec-ch-ua-mobile" => array:1 [ 0 => "?0" ] "sec-ch-ua" => array:1 [ 0 => ""HeadlessChrome";v="129", "Not=A?Brand";v="8", "Chromium";v="129"" ] "cache-control" => array:1 [ 0 => "no-cache" ] "pragma" => array:1 [ 0 => "no-cache" ] "x-amzn-trace-id" => array:1 [ 0 => "Root=1-68000b88-1ae2201a5f0f58fc1def4ea5" ] "host" => array:1 [ 0 => "staging.dev.24houranswers.com" ] "x-forwarded-port" => array:1 [ 0 => "443" ] "x-forwarded-proto" => array:1 [ 0 => "https" ] "x-forwarded-for" => array:1 [ 0 => "18.189.28.82" ] "content-length" => array:1 [ 0 => "" ] "content-type" => array:1 [ 0 => "" ] ]
          request_cookies
          []
          
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