Which corrections should I use? T-test for differences in means with different sample sizes and standard deviations

The name of the pictureThe name of the pictureThe name of the pictureClash Royale CLAN TAG#URR8PPP












3












$begingroup$


I have two samples, coming from different populations.
One sample has 8,000 records, a mean of 5 and a sd of 0.5
The second has 1,500 records, a mean of 7 and a sd of 1.5
The distributions are close to normal.



This is coming from the behaviour of two kind of devices, and I want to understand if the output of one is of higher quality than the other.



Can I apply a $t$-test here? What cautions should I have or which corrections/alternative test do I have?










share|cite|improve this question











$endgroup$
















    3












    $begingroup$


    I have two samples, coming from different populations.
    One sample has 8,000 records, a mean of 5 and a sd of 0.5
    The second has 1,500 records, a mean of 7 and a sd of 1.5
    The distributions are close to normal.



    This is coming from the behaviour of two kind of devices, and I want to understand if the output of one is of higher quality than the other.



    Can I apply a $t$-test here? What cautions should I have or which corrections/alternative test do I have?










    share|cite|improve this question











    $endgroup$














      3












      3








      3





      $begingroup$


      I have two samples, coming from different populations.
      One sample has 8,000 records, a mean of 5 and a sd of 0.5
      The second has 1,500 records, a mean of 7 and a sd of 1.5
      The distributions are close to normal.



      This is coming from the behaviour of two kind of devices, and I want to understand if the output of one is of higher quality than the other.



      Can I apply a $t$-test here? What cautions should I have or which corrections/alternative test do I have?










      share|cite|improve this question











      $endgroup$




      I have two samples, coming from different populations.
      One sample has 8,000 records, a mean of 5 and a sd of 0.5
      The second has 1,500 records, a mean of 7 and a sd of 1.5
      The distributions are close to normal.



      This is coming from the behaviour of two kind of devices, and I want to understand if the output of one is of higher quality than the other.



      Can I apply a $t$-test here? What cautions should I have or which corrections/alternative test do I have?







      statistical-significance t-test inference






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Feb 17 at 16:26









      StatsStudent

      6,01332044




      6,01332044










      asked Feb 16 at 14:00









      LuisLuis

      85119




      85119




















          2 Answers
          2






          active

          oldest

          votes


















          2












          $begingroup$

          Assuming your samples are independent, then Welch's t-test does seem to be appropriate here, since it appears you have unequal variances (but you can formally test this too if you want through Levene's Test for Equality of Variances).



          That being said, since you have quite large samples from both device 1 and device 2, then you can appeal to the central limit theorem and use:



          begineqnarray*
          Z & = & fracbarX-barYsqrtfracs_1^2n_1+fracs_2^2n_2sim N(0,1)\
          endeqnarray*



          under the null hypothesis of equal means. Here, $barX$ and $barY$ and sample means from device 1 and device 2, respectively and $s_i^2$ and $n_i$ are the sample variance and sample sizes from the ith device $i=1,2$. Note that in large sample inference, you don't need to concern yourself with unequal variances.



          Then a 95% confidence interval for your estimate would be given by:



          begineqnarray*
          barX-barY & pm & Z_alpha/2sqrtfracs_1^2n_1+fracs_2^2n_2
          endeqnarray*



          where $Z_alpha/2$ is the upper $alpha/2$ point of the standard normal distribution.



          All this being said, I wholeheartedly agree with the answer provided by Stefan. These sample sizes are really large and he's provided sound advice that you should follow. You should focus on what is an important practical difference. Is a 0.0001 mean difference between device 1 and device 2 important to you? Is it still important if device 1 costs three times as much as device 2?






          share|cite|improve this answer











          $endgroup$




















            2












            $begingroup$

            With such a huge sample size almost any slight differences in those two means will be declared significant. Instead, I would try to visualize your samples in different ways to learn more about the shape of the data.



            Also how is "higher quality" defined by you? Does it mean that the mean outcomes should be different? Or does it perhaps apply more to the variances between the samples, e.g. less variation more desirable?



            Here are some ideas how to visualize the data using R:



            require(ggplot2)
            require(gridExtra)

            d1 <- data.frame(Y = rnorm(8000, 5, 0.5), X = "A")
            d2 <- data.frame(Y = rnorm(1500, 7, 1.5), X = "B")
            d <- rbind(d1, d2)

            p1 <- ggplot(d, aes(Y, group = X)) + geom_density() + ggtitle("Density plot")
            p2 <- ggplot(d, aes(X, Y)) + geom_boxplot() + ggtitle("Boxplot")
            p3 <- ggplot(d, aes(X, Y)) + geom_violin() + ggtitle("Violin plot")

            grid.arrange(p1, p2, p3, ncol = 1)


            enter image description here






            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: "65"
              ;
              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: false,
              noModals: true,
              showLowRepImageUploadWarning: true,
              reputationToPostImages: null,
              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
              ,
              onDemand: true,
              discardSelector: ".discard-answer"
              ,immediatelyShowMarkdownHelp:true
              );



              );













              draft saved

              draft discarded


















              StackExchange.ready(
              function ()
              StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fstats.stackexchange.com%2fquestions%2f392823%2fwhich-corrections-should-i-use-t-test-for-differences-in-means-with-different-s%23new-answer', 'question_page');

              );

              Post as a guest















              Required, but never shown

























              2 Answers
              2






              active

              oldest

              votes








              2 Answers
              2






              active

              oldest

              votes









              active

              oldest

              votes






              active

              oldest

              votes









              2












              $begingroup$

              Assuming your samples are independent, then Welch's t-test does seem to be appropriate here, since it appears you have unequal variances (but you can formally test this too if you want through Levene's Test for Equality of Variances).



              That being said, since you have quite large samples from both device 1 and device 2, then you can appeal to the central limit theorem and use:



              begineqnarray*
              Z & = & fracbarX-barYsqrtfracs_1^2n_1+fracs_2^2n_2sim N(0,1)\
              endeqnarray*



              under the null hypothesis of equal means. Here, $barX$ and $barY$ and sample means from device 1 and device 2, respectively and $s_i^2$ and $n_i$ are the sample variance and sample sizes from the ith device $i=1,2$. Note that in large sample inference, you don't need to concern yourself with unequal variances.



              Then a 95% confidence interval for your estimate would be given by:



              begineqnarray*
              barX-barY & pm & Z_alpha/2sqrtfracs_1^2n_1+fracs_2^2n_2
              endeqnarray*



              where $Z_alpha/2$ is the upper $alpha/2$ point of the standard normal distribution.



              All this being said, I wholeheartedly agree with the answer provided by Stefan. These sample sizes are really large and he's provided sound advice that you should follow. You should focus on what is an important practical difference. Is a 0.0001 mean difference between device 1 and device 2 important to you? Is it still important if device 1 costs three times as much as device 2?






              share|cite|improve this answer











              $endgroup$

















                2












                $begingroup$

                Assuming your samples are independent, then Welch's t-test does seem to be appropriate here, since it appears you have unequal variances (but you can formally test this too if you want through Levene's Test for Equality of Variances).



                That being said, since you have quite large samples from both device 1 and device 2, then you can appeal to the central limit theorem and use:



                begineqnarray*
                Z & = & fracbarX-barYsqrtfracs_1^2n_1+fracs_2^2n_2sim N(0,1)\
                endeqnarray*



                under the null hypothesis of equal means. Here, $barX$ and $barY$ and sample means from device 1 and device 2, respectively and $s_i^2$ and $n_i$ are the sample variance and sample sizes from the ith device $i=1,2$. Note that in large sample inference, you don't need to concern yourself with unequal variances.



                Then a 95% confidence interval for your estimate would be given by:



                begineqnarray*
                barX-barY & pm & Z_alpha/2sqrtfracs_1^2n_1+fracs_2^2n_2
                endeqnarray*



                where $Z_alpha/2$ is the upper $alpha/2$ point of the standard normal distribution.



                All this being said, I wholeheartedly agree with the answer provided by Stefan. These sample sizes are really large and he's provided sound advice that you should follow. You should focus on what is an important practical difference. Is a 0.0001 mean difference between device 1 and device 2 important to you? Is it still important if device 1 costs three times as much as device 2?






                share|cite|improve this answer











                $endgroup$















                  2












                  2








                  2





                  $begingroup$

                  Assuming your samples are independent, then Welch's t-test does seem to be appropriate here, since it appears you have unequal variances (but you can formally test this too if you want through Levene's Test for Equality of Variances).



                  That being said, since you have quite large samples from both device 1 and device 2, then you can appeal to the central limit theorem and use:



                  begineqnarray*
                  Z & = & fracbarX-barYsqrtfracs_1^2n_1+fracs_2^2n_2sim N(0,1)\
                  endeqnarray*



                  under the null hypothesis of equal means. Here, $barX$ and $barY$ and sample means from device 1 and device 2, respectively and $s_i^2$ and $n_i$ are the sample variance and sample sizes from the ith device $i=1,2$. Note that in large sample inference, you don't need to concern yourself with unequal variances.



                  Then a 95% confidence interval for your estimate would be given by:



                  begineqnarray*
                  barX-barY & pm & Z_alpha/2sqrtfracs_1^2n_1+fracs_2^2n_2
                  endeqnarray*



                  where $Z_alpha/2$ is the upper $alpha/2$ point of the standard normal distribution.



                  All this being said, I wholeheartedly agree with the answer provided by Stefan. These sample sizes are really large and he's provided sound advice that you should follow. You should focus on what is an important practical difference. Is a 0.0001 mean difference between device 1 and device 2 important to you? Is it still important if device 1 costs three times as much as device 2?






                  share|cite|improve this answer











                  $endgroup$



                  Assuming your samples are independent, then Welch's t-test does seem to be appropriate here, since it appears you have unequal variances (but you can formally test this too if you want through Levene's Test for Equality of Variances).



                  That being said, since you have quite large samples from both device 1 and device 2, then you can appeal to the central limit theorem and use:



                  begineqnarray*
                  Z & = & fracbarX-barYsqrtfracs_1^2n_1+fracs_2^2n_2sim N(0,1)\
                  endeqnarray*



                  under the null hypothesis of equal means. Here, $barX$ and $barY$ and sample means from device 1 and device 2, respectively and $s_i^2$ and $n_i$ are the sample variance and sample sizes from the ith device $i=1,2$. Note that in large sample inference, you don't need to concern yourself with unequal variances.



                  Then a 95% confidence interval for your estimate would be given by:



                  begineqnarray*
                  barX-barY & pm & Z_alpha/2sqrtfracs_1^2n_1+fracs_2^2n_2
                  endeqnarray*



                  where $Z_alpha/2$ is the upper $alpha/2$ point of the standard normal distribution.



                  All this being said, I wholeheartedly agree with the answer provided by Stefan. These sample sizes are really large and he's provided sound advice that you should follow. You should focus on what is an important practical difference. Is a 0.0001 mean difference between device 1 and device 2 important to you? Is it still important if device 1 costs three times as much as device 2?







                  share|cite|improve this answer














                  share|cite|improve this answer



                  share|cite|improve this answer








                  edited Feb 16 at 15:37

























                  answered Feb 16 at 15:13









                  StatsStudentStatsStudent

                  6,01332044




                  6,01332044























                      2












                      $begingroup$

                      With such a huge sample size almost any slight differences in those two means will be declared significant. Instead, I would try to visualize your samples in different ways to learn more about the shape of the data.



                      Also how is "higher quality" defined by you? Does it mean that the mean outcomes should be different? Or does it perhaps apply more to the variances between the samples, e.g. less variation more desirable?



                      Here are some ideas how to visualize the data using R:



                      require(ggplot2)
                      require(gridExtra)

                      d1 <- data.frame(Y = rnorm(8000, 5, 0.5), X = "A")
                      d2 <- data.frame(Y = rnorm(1500, 7, 1.5), X = "B")
                      d <- rbind(d1, d2)

                      p1 <- ggplot(d, aes(Y, group = X)) + geom_density() + ggtitle("Density plot")
                      p2 <- ggplot(d, aes(X, Y)) + geom_boxplot() + ggtitle("Boxplot")
                      p3 <- ggplot(d, aes(X, Y)) + geom_violin() + ggtitle("Violin plot")

                      grid.arrange(p1, p2, p3, ncol = 1)


                      enter image description here






                      share|cite|improve this answer











                      $endgroup$

















                        2












                        $begingroup$

                        With such a huge sample size almost any slight differences in those two means will be declared significant. Instead, I would try to visualize your samples in different ways to learn more about the shape of the data.



                        Also how is "higher quality" defined by you? Does it mean that the mean outcomes should be different? Or does it perhaps apply more to the variances between the samples, e.g. less variation more desirable?



                        Here are some ideas how to visualize the data using R:



                        require(ggplot2)
                        require(gridExtra)

                        d1 <- data.frame(Y = rnorm(8000, 5, 0.5), X = "A")
                        d2 <- data.frame(Y = rnorm(1500, 7, 1.5), X = "B")
                        d <- rbind(d1, d2)

                        p1 <- ggplot(d, aes(Y, group = X)) + geom_density() + ggtitle("Density plot")
                        p2 <- ggplot(d, aes(X, Y)) + geom_boxplot() + ggtitle("Boxplot")
                        p3 <- ggplot(d, aes(X, Y)) + geom_violin() + ggtitle("Violin plot")

                        grid.arrange(p1, p2, p3, ncol = 1)


                        enter image description here






                        share|cite|improve this answer











                        $endgroup$















                          2












                          2








                          2





                          $begingroup$

                          With such a huge sample size almost any slight differences in those two means will be declared significant. Instead, I would try to visualize your samples in different ways to learn more about the shape of the data.



                          Also how is "higher quality" defined by you? Does it mean that the mean outcomes should be different? Or does it perhaps apply more to the variances between the samples, e.g. less variation more desirable?



                          Here are some ideas how to visualize the data using R:



                          require(ggplot2)
                          require(gridExtra)

                          d1 <- data.frame(Y = rnorm(8000, 5, 0.5), X = "A")
                          d2 <- data.frame(Y = rnorm(1500, 7, 1.5), X = "B")
                          d <- rbind(d1, d2)

                          p1 <- ggplot(d, aes(Y, group = X)) + geom_density() + ggtitle("Density plot")
                          p2 <- ggplot(d, aes(X, Y)) + geom_boxplot() + ggtitle("Boxplot")
                          p3 <- ggplot(d, aes(X, Y)) + geom_violin() + ggtitle("Violin plot")

                          grid.arrange(p1, p2, p3, ncol = 1)


                          enter image description here






                          share|cite|improve this answer











                          $endgroup$



                          With such a huge sample size almost any slight differences in those two means will be declared significant. Instead, I would try to visualize your samples in different ways to learn more about the shape of the data.



                          Also how is "higher quality" defined by you? Does it mean that the mean outcomes should be different? Or does it perhaps apply more to the variances between the samples, e.g. less variation more desirable?



                          Here are some ideas how to visualize the data using R:



                          require(ggplot2)
                          require(gridExtra)

                          d1 <- data.frame(Y = rnorm(8000, 5, 0.5), X = "A")
                          d2 <- data.frame(Y = rnorm(1500, 7, 1.5), X = "B")
                          d <- rbind(d1, d2)

                          p1 <- ggplot(d, aes(Y, group = X)) + geom_density() + ggtitle("Density plot")
                          p2 <- ggplot(d, aes(X, Y)) + geom_boxplot() + ggtitle("Boxplot")
                          p3 <- ggplot(d, aes(X, Y)) + geom_violin() + ggtitle("Violin plot")

                          grid.arrange(p1, p2, p3, ncol = 1)


                          enter image description here







                          share|cite|improve this answer














                          share|cite|improve this answer



                          share|cite|improve this answer








                          edited Feb 19 at 14:53

























                          answered Feb 16 at 15:20









                          StefanStefan

                          3,5811931




                          3,5811931



























                              draft saved

                              draft discarded
















































                              Thanks for contributing an answer to Cross Validated!


                              • 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%2fstats.stackexchange.com%2fquestions%2f392823%2fwhich-corrections-should-i-use-t-test-for-differences-in-means-with-different-s%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

                              How to check contact read email or not when send email to Individual?

                              Displaying single band from multi-band raster using QGIS

                              How many registers does an x86_64 CPU actually have?