Type 2 Error Question - How to calculate for a two tailed?












1












$begingroup$


The modulus of rupture (MOR) for a particular grade of pencil lead is known to have
a standard deviation of 250 psi. Process standards call for a target value of 6500 psi
for the true mean MOR. For each batch, an inspector tests a random sample of 16
leads. Management wishes to detect any change in the true mean MOR. (Assume normal
distribution.)



QUESTION : Find the probability of type II error of the test when the true mean MOR is 6400.



How do we solve for two tails? i am able to get 0.4821 but with 1 tail method.
6397-6400/62.5 = 0.045
than using normcdf on MATLAB, i got this value






probability statistics data-analysis hypothesis-testing







share|cite|improve this question









$endgroup$

















    1












    $begingroup$


    The modulus of rupture (MOR) for a particular grade of pencil lead is known to have
    a standard deviation of 250 psi. Process standards call for a target value of 6500 psi
    for the true mean MOR. For each batch, an inspector tests a random sample of 16
    leads. Management wishes to detect any change in the true mean MOR. (Assume normal
    distribution.)



    QUESTION : Find the probability of type II error of the test when the true mean MOR is 6400.



    How do we solve for two tails? i am able to get 0.4821 but with 1 tail method.
    6397-6400/62.5 = 0.045
    than using normcdf on MATLAB, i got this value






    probability statistics data-analysis hypothesis-testing







    share|cite|improve this question









    $endgroup$















      1












      1








      1





      $begingroup$


      The modulus of rupture (MOR) for a particular grade of pencil lead is known to have
      a standard deviation of 250 psi. Process standards call for a target value of 6500 psi
      for the true mean MOR. For each batch, an inspector tests a random sample of 16
      leads. Management wishes to detect any change in the true mean MOR. (Assume normal
      distribution.)



      QUESTION : Find the probability of type II error of the test when the true mean MOR is 6400.



      How do we solve for two tails? i am able to get 0.4821 but with 1 tail method.
      6397-6400/62.5 = 0.045
      than using normcdf on MATLAB, i got this value






      probability statistics data-analysis hypothesis-testing







      share|cite|improve this question









      $endgroup$




      The modulus of rupture (MOR) for a particular grade of pencil lead is known to have
      a standard deviation of 250 psi. Process standards call for a target value of 6500 psi
      for the true mean MOR. For each batch, an inspector tests a random sample of 16
      leads. Management wishes to detect any change in the true mean MOR. (Assume normal
      distribution.)



      QUESTION : Find the probability of type II error of the test when the true mean MOR is 6400.



      How do we solve for two tails? i am able to get 0.4821 but with 1 tail method.
      6397-6400/62.5 = 0.045
      than using normcdf on MATLAB, i got this value






      probability statistics data-analysis hypothesis-testing




      probability statistics data-analysis hypothesis-testing






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Apr 27 '16 at 1:12









      user334902user334902

      115




      115






















          1 Answer
          1






          active

          oldest

          votes


















          0












          $begingroup$

          You have not said what significance level you are using, but it seems that what you did for one tail used $5%$ and




          • standard error of the mean $dfrac{250}{sqrt{16}} = 62.5$

          • If the true population mean is $6500$ then there is a $5%$ probability that the the sample mean will be below $6500 + 62.5 Phi^{-1}(0.05) approx 6397$

          • If the true population mean is $6400$ then there is about a $Phileft(dfrac{6397-6400}{62.5}right) approx 48.21%$ probability that the the sample mean will be below $6397$


          You need to do something similar for two tails:




          • If the true population mean is $6500$ then there is a $2.5%$ probability that the the sample mean will be below $6500 + 62.5 Phi^{-1}(0.025) approx 6377.5$

          • If the true population mean is $6500$ then there is a $2.5%$ probability that the the sample mean will be above $6500 + 62.5 Phi^{-1}(0.975) approx 6622.5$

          • If the true population mean is $6400$ then there is about a $Phileft(dfrac{6377.5-6400}{62.5}right) approx 35.94%$ probability that the the sample mean will be below $6377.5$

          • If the true population mean is $6400$ then there is about a $1-Phileft(dfrac{6622.5-6400}{62.5}right) approx 0.02%$ probability that the the sample mean will be above $6622.5$

          • So the probability of a type II error of the test when the true mean is $6400$ is about $35.96%$






          share|cite|improve this answer









          $endgroup$













            Your Answer





            StackExchange.ifUsing("editor", function () {
            return StackExchange.using("mathjaxEditing", function () {
            StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
            StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
            });
            });
            }, "mathjax-editing");

            StackExchange.ready(function() {
            var channelOptions = {
            tags: "".split(" "),
            id: "69"
            };
            initTagRenderer("".split(" "), "".split(" "), channelOptions);

            StackExchange.using("externalEditor", function() {
            // Have to fire editor after snippets, if snippets enabled
            if (StackExchange.settings.snippets.snippetsEnabled) {
            StackExchange.using("snippets", function() {
            createEditor();
            });
            }
            else {
            createEditor();
            }
            });

            function createEditor() {
            StackExchange.prepareEditor({
            heartbeatType: 'answer',
            autoActivateHeartbeat: false,
            convertImagesToLinks: true,
            noModals: true,
            showLowRepImageUploadWarning: true,
            reputationToPostImages: 10,
            bindNavPrevention: true,
            postfix: "",
            imageUploader: {
            brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
            contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
            allowUrls: true
            },
            noCode: true, onDemand: true,
            discardSelector: ".discard-answer"
            ,immediatelyShowMarkdownHelp:true
            });


            }
            });














            draft saved

            draft discarded


















            StackExchange.ready(
            function () {
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f1760432%2ftype-2-error-question-how-to-calculate-for-a-two-tailed%23new-answer', 'question_page');
            }
            );

            Post as a guest















            Required, but never shown

























            1 Answer
            1






            active

            oldest

            votes








            1 Answer
            1






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes









            0












            $begingroup$

            You have not said what significance level you are using, but it seems that what you did for one tail used $5%$ and




            • standard error of the mean $dfrac{250}{sqrt{16}} = 62.5$

            • If the true population mean is $6500$ then there is a $5%$ probability that the the sample mean will be below $6500 + 62.5 Phi^{-1}(0.05) approx 6397$

            • If the true population mean is $6400$ then there is about a $Phileft(dfrac{6397-6400}{62.5}right) approx 48.21%$ probability that the the sample mean will be below $6397$


            You need to do something similar for two tails:




            • If the true population mean is $6500$ then there is a $2.5%$ probability that the the sample mean will be below $6500 + 62.5 Phi^{-1}(0.025) approx 6377.5$

            • If the true population mean is $6500$ then there is a $2.5%$ probability that the the sample mean will be above $6500 + 62.5 Phi^{-1}(0.975) approx 6622.5$

            • If the true population mean is $6400$ then there is about a $Phileft(dfrac{6377.5-6400}{62.5}right) approx 35.94%$ probability that the the sample mean will be below $6377.5$

            • If the true population mean is $6400$ then there is about a $1-Phileft(dfrac{6622.5-6400}{62.5}right) approx 0.02%$ probability that the the sample mean will be above $6622.5$

            • So the probability of a type II error of the test when the true mean is $6400$ is about $35.96%$






            share|cite|improve this answer









            $endgroup$


















              0












              $begingroup$

              You have not said what significance level you are using, but it seems that what you did for one tail used $5%$ and




              • standard error of the mean $dfrac{250}{sqrt{16}} = 62.5$

              • If the true population mean is $6500$ then there is a $5%$ probability that the the sample mean will be below $6500 + 62.5 Phi^{-1}(0.05) approx 6397$

              • If the true population mean is $6400$ then there is about a $Phileft(dfrac{6397-6400}{62.5}right) approx 48.21%$ probability that the the sample mean will be below $6397$


              You need to do something similar for two tails:




              • If the true population mean is $6500$ then there is a $2.5%$ probability that the the sample mean will be below $6500 + 62.5 Phi^{-1}(0.025) approx 6377.5$

              • If the true population mean is $6500$ then there is a $2.5%$ probability that the the sample mean will be above $6500 + 62.5 Phi^{-1}(0.975) approx 6622.5$

              • If the true population mean is $6400$ then there is about a $Phileft(dfrac{6377.5-6400}{62.5}right) approx 35.94%$ probability that the the sample mean will be below $6377.5$

              • If the true population mean is $6400$ then there is about a $1-Phileft(dfrac{6622.5-6400}{62.5}right) approx 0.02%$ probability that the the sample mean will be above $6622.5$

              • So the probability of a type II error of the test when the true mean is $6400$ is about $35.96%$






              share|cite|improve this answer









              $endgroup$
















                0












                0








                0





                $begingroup$

                You have not said what significance level you are using, but it seems that what you did for one tail used $5%$ and




                • standard error of the mean $dfrac{250}{sqrt{16}} = 62.5$

                • If the true population mean is $6500$ then there is a $5%$ probability that the the sample mean will be below $6500 + 62.5 Phi^{-1}(0.05) approx 6397$

                • If the true population mean is $6400$ then there is about a $Phileft(dfrac{6397-6400}{62.5}right) approx 48.21%$ probability that the the sample mean will be below $6397$


                You need to do something similar for two tails:




                • If the true population mean is $6500$ then there is a $2.5%$ probability that the the sample mean will be below $6500 + 62.5 Phi^{-1}(0.025) approx 6377.5$

                • If the true population mean is $6500$ then there is a $2.5%$ probability that the the sample mean will be above $6500 + 62.5 Phi^{-1}(0.975) approx 6622.5$

                • If the true population mean is $6400$ then there is about a $Phileft(dfrac{6377.5-6400}{62.5}right) approx 35.94%$ probability that the the sample mean will be below $6377.5$

                • If the true population mean is $6400$ then there is about a $1-Phileft(dfrac{6622.5-6400}{62.5}right) approx 0.02%$ probability that the the sample mean will be above $6622.5$

                • So the probability of a type II error of the test when the true mean is $6400$ is about $35.96%$






                share|cite|improve this answer









                $endgroup$



                You have not said what significance level you are using, but it seems that what you did for one tail used $5%$ and




                • standard error of the mean $dfrac{250}{sqrt{16}} = 62.5$

                • If the true population mean is $6500$ then there is a $5%$ probability that the the sample mean will be below $6500 + 62.5 Phi^{-1}(0.05) approx 6397$

                • If the true population mean is $6400$ then there is about a $Phileft(dfrac{6397-6400}{62.5}right) approx 48.21%$ probability that the the sample mean will be below $6397$


                You need to do something similar for two tails:




                • If the true population mean is $6500$ then there is a $2.5%$ probability that the the sample mean will be below $6500 + 62.5 Phi^{-1}(0.025) approx 6377.5$

                • If the true population mean is $6500$ then there is a $2.5%$ probability that the the sample mean will be above $6500 + 62.5 Phi^{-1}(0.975) approx 6622.5$

                • If the true population mean is $6400$ then there is about a $Phileft(dfrac{6377.5-6400}{62.5}right) approx 35.94%$ probability that the the sample mean will be below $6377.5$

                • If the true population mean is $6400$ then there is about a $1-Phileft(dfrac{6622.5-6400}{62.5}right) approx 0.02%$ probability that the the sample mean will be above $6622.5$

                • So the probability of a type II error of the test when the true mean is $6400$ is about $35.96%$







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Apr 28 '16 at 7:59









                HenryHenry

                101k481168




                101k481168






























                    draft saved

                    draft discarded




















































                    Thanks for contributing an answer to Mathematics Stack Exchange!


                    • Please be sure to answer the question. Provide details and share your research!

                    But avoid …



                    • Asking for help, clarification, or responding to other answers.

                    • Making statements based on opinion; back them up with references or personal experience.


                    Use MathJax to format equations. MathJax reference.


                    To learn more, see our tips on writing great answers.




                    draft saved


                    draft discarded














                    StackExchange.ready(
                    function () {
                    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f1760432%2ftype-2-error-question-how-to-calculate-for-a-two-tailed%23new-answer', 'question_page');
                    }
                    );

                    Post as a guest















                    Required, but never shown





















































                    Required, but never shown














                    Required, but never shown












                    Required, but never shown







                    Required, but never shown

































                    Required, but never shown














                    Required, but never shown












                    Required, but never shown







                    Required, but never shown







                    -

                    Popular posts from this blog

                    mysqli_query(): Empty query in /home/lucindabrummitt/public_html/blog/wp-includes/wp-db.php on line 1924

                    Image
                    0 I just uploaded a new plugin to my wordpress site and I got this warning: mysqli_query(): Empty query in /home/lucindabrummitt/public_html/blog/wp-includes/wp-db.php on line 1924 Usually if I have a problem, I can find answers on google but this one seems pretty particular and I'm not finding any info on it except for this forum. I'm not terribly techie but if someone can tell me how to read this and how to change it I can follow directions pretty well. Thanks All! PS If I don't have the proper tags here could someone add the appropriate tags? php mysql wordpress share | improve this question asked Nov 16 '18 at 20:41 ...
                    Read more

                    How to change which sound is reproduced for terminal bell?

                    Image
                    3 Recently upgrade to 18.10 from 18.04, and using the terminal is becoming annoying: the new alert sound is a drip ( /usr/share/sounds/gnome/default/alerts/drip.ogg ), and the previous one is still located at ( /usr/share/sounds/ubuntu/stereo/bell.ogg ). My issue is that at the gnome-terminal, when autocompleting or pressing left BOTH sounds are reproduced, at the same time. I want to just have ubuntu/stereo/bell.ogg active, not both . At the moment I have been only able to reduce the volume of the alerts (but still both are played), or turn the alerts down (but I dislike both "solutions"). gnome-terminal 18.10 teminal-bell share | improve this question ...
                    Read more

                    Can I use Tabulator js library in my java Spring + Thymeleaf project?

                    Image
                    0 I want to add Tabulator js to my java project. I added tabulator.min.js, tabulator.min.css to resources folder. After that added to my html page these dependencies as recommending in readme.md. <link rel="stylesheet" th:href="@{/css/tabulator.min.css}"> <script type="text/javascript" th:src="@{/js/tabulator.min.js}"></script> Now my tables view all elements, that I have in the html page, but not correct. In the documentation of Tabulator is example how to create header and table, var table = new Tabulator("#example-table", { columns:[ {title:"Name", field:"name", sorter:"string", width:200, editor:true}, {title:"Age", field:"age", sorter:"...
                    Read more