2 Fixed obvious typos edited Feb 24 '12 at 18:33 whuber♦ 213k3434 gold badges468468 silver badges856856 bronze badges A lot of this comes down to what question you are actually asking, how you design your study, and even what you mean by equal. I ran accrossaccros an interesting little insert in the British Medical Journal once that talked about what people interpreted certain phases to mean. It turns out that "always" can mean that something happens as low as 91% of the time (BMJ VOLUME 333 26 AUGUST 2006 page 445). So maybe equal and equivalent (or within X% for some value of X) could be thought to mean the same thing. And lets ask the computer a simple equality, using R: > (1e+5 + 1e-50) == (1e+5 - 1e-50) [1] TRUE  Now a pure mathematician using infinite precision might say that those 2 values are not equal, but R says they are and for most practical cases they would be (If you offereredoffered to give me $$(1e+5 + 1e-50), but the amount ended up being$$$$\$$(1e+5 + 1e-50), but the amount ended up being $$\$$(1e+5 - 1e-50) I would not refuse the money because it differed from what was prommisedpromised). Further if our alternative hypothesis is $$H_a: \mu > \mu_0$$ we often write the null as $$H_0: \mu=\mu_0$$ even though technically the real null is $$H_0: \mu \le \mu_0$$, but we work with the equality as null since if we can show that $$\mu$$ is bigger than $$\mu_0$$ then we also know that it is bigger than all the values less than $$\mu_0$$. And isn't a two-tailed test really just 2 one-tailed tests? After all, would you really say that $$\mu \ne \mu_0$$ but refuse to say which side of $$\mu_0$$ $$\mu$$ is on? This is partly why there is a trend towards using confidence intervals in place of p-values when possible, if my confidence interval for $$\mu$$ includes $$\mu_0$$ then while I may not be willing to believe that $$\mu$$ is exactly equal to $$\mu_0$$, I cannot say for certain which side of $$\mu_0$$ $$\um$$$$\mu$$ lies on, which means they might as well be equal for practical purposes. A lot of this comes down to asking the right question and desigingdesigning the right study for that question. If you end up with enough data to show that a practically meaningless difference is statistically significant, then you have wasted resources getting that much data. It would have been better to decide what a meaningful difference would be and designed the study to give you enough power to detect that difference but not smaller. And if we really want to split hairs, how do we define what parts of the lamb are on the right and which are on the left? If we define it by a line that by definition has equal number of hairs on each side then the answer to the above question becomes "Of Course it is". A lot of this comes down to what question you are actually asking, how you design your study, and even what you mean by equal. I ran accross an interesting little insert in the British Medical Journal once that talked about what people interpreted certain phases to mean. It turns out that "always" can mean that something happens as low as 91% of the time (BMJ VOLUME 333 26 AUGUST 2006 page 445). So maybe equal and equivalent (or within X% for some value of X) could be thought to mean the same thing. And lets ask the computer a simple equality, using R: > (1e+5 + 1e-50) == (1e+5 - 1e-50) [1] TRUE  Now a pure mathematician using infinite precision might say that those 2 values are not equal, but R says they are and for most practical cases they would be (If you offerered to give me $$(1e+5 + 1e-50), but the amount ended up being$$(1e+5 - 1e-50) I would not refuse the money because it differed from what was prommised). Further if our alternative hypothesis is $$H_a: \mu > \mu_0$$ we often write the null as $$H_0: \mu=\mu_0$$ even though technically the real null is $$H_0: \mu \le \mu_0$$, but we work with the equality as null since if we can show that $$\mu$$ is bigger than $$\mu_0$$ then we also know that it is bigger than all the values less than $$\mu_0$$. And isn't a two-tailed test really just 2 one-tailed tests? After all, would you really say that $$\mu \ne \mu_0$$ but refuse to say which side of $$\mu_0$$ $$\mu$$ is on? This is partly why there is a trend towards using confidence intervals in place of p-values when possible, if my confidence interval for $$\mu$$ includes $$\mu_0$$ then while I may not be willing to believe that $$\mu$$ is exactly equal to $$\mu_0$$, I cannot say for certain which side of $$\mu_0$$ $$\um$$ lies on, which means they might as well be equal for practical purposes. A lot of this comes down to asking the right question and desiging the right study for that question. If you end up with enough data to show that a practically meaningless difference is statistically significant, then you have wasted resources getting that much data. It would have been better to decide what a meaningful difference would be and designed the study to give you enough power to detect that difference but not smaller. And if we really want to split hairs, how do we define what parts of the lamb are on the right and which are on the left? If we define it by a line that by definition has equal number of hairs on each side then the answer to the above question becomes "Of Course it is". A lot of this comes down to what question you are actually asking, how you design your study, and even what you mean by equal. I ran accros an interesting little insert in the British Medical Journal once that talked about what people interpreted certain phases to mean. It turns out that "always" can mean that something happens as low as 91% of the time (BMJ VOLUME 333 26 AUGUST 2006 page 445). So maybe equal and equivalent (or within X% for some value of X) could be thought to mean the same thing. And lets ask the computer a simple equality, using R: > (1e+5 + 1e-50) == (1e+5 - 1e-50) [1] TRUE  Now a pure mathematician using infinite precision might say that those 2 values are not equal, but R says they are and for most practical cases they would be (If you offered to give me $$\$$(1e+5 + 1e-50), but the amount ended up being $$\$$(1e+5 - 1e-50) I would not refuse the money because it differed from what was promised). Further if our alternative hypothesis is $$H_a: \mu > \mu_0$$ we often write the null as $$H_0: \mu=\mu_0$$ even though technically the real null is $$H_0: \mu \le \mu_0$$, but we work with the equality as null since if we can show that $$\mu$$ is bigger than $$\mu_0$$ then we also know that it is bigger than all the values less than $$\mu_0$$. And isn't a two-tailed test really just 2 one-tailed tests? After all, would you really say that $$\mu \ne \mu_0$$ but refuse to say which side of $$\mu_0$$ $$\mu$$ is on? This is partly why there is a trend towards using confidence intervals in place of p-values when possible, if my confidence interval for $$\mu$$ includes $$\mu_0$$ then while I may not be willing to believe that $$\mu$$ is exactly equal to $$\mu_0$$, I cannot say for certain which side of $$\mu_0$$ $$\mu$$ lies on, which means they might as well be equal for practical purposes. A lot of this comes down to asking the right question and designing the right study for that question. If you end up with enough data to show that a practically meaningless difference is statistically significant, then you have wasted resources getting that much data. It would have been better to decide what a meaningful difference would be and designed the study to give you enough power to detect that difference but not smaller. And if we really want to split hairs, how do we define what parts of the lamb are on the right and which are on the left? If we define it by a line that by definition has equal number of hairs on each side then the answer to the above question becomes "Of Course it is". 1 answered Feb 24 '12 at 18:25 Greg Snow 40.6k11 gold badge6969 silver badges134134 bronze badges A lot of this comes down to what question you are actually asking, how you design your study, and even what you mean by equal. I ran accross an interesting little insert in the British Medical Journal once that talked about what people interpreted certain phases to mean. It turns out that "always" can mean that something happens as low as 91% of the time (BMJ VOLUME 333 26 AUGUST 2006 page 445). So maybe equal and equivalent (or within X% for some value of X) could be thought to mean the same thing. And lets ask the computer a simple equality, using R: > (1e+5 + 1e-50) == (1e+5 - 1e-50) [1] TRUE  Now a pure mathematician using infinite precision might say that those 2 values are not equal, but R says they are and for most practical cases they would be (If you offerered to give me $$(1e+5 + 1e-50), but the amount ended up being$$(1e+5 - 1e-50) I would not refuse the money because it differed from what was prommised). Further if our alternative hypothesis is $$H_a: \mu > \mu_0$$ we often write the null as $$H_0: \mu=\mu_0$$ even though technically the real null is $$H_0: \mu \le \mu_0$$, but we work with the equality as null since if we can show that $$\mu$$ is bigger than $$\mu_0$$ then we also know that it is bigger than all the values less than $$\mu_0$$. And isn't a two-tailed test really just 2 one-tailed tests? After all, would you really say that $$\mu \ne \mu_0$$ but refuse to say which side of $$\mu_0$$ $$\mu$$ is on? This is partly why there is a trend towards using confidence intervals in place of p-values when possible, if my confidence interval for $$\mu$$ includes $$\mu_0$$ then while I may not be willing to believe that $$\mu$$ is exactly equal to $$\mu_0$$, I cannot say for certain which side of $$\mu_0$$ $$\um$$ lies on, which means they might as well be equal for practical purposes. A lot of this comes down to asking the right question and desiging the right study for that question. If you end up with enough data to show that a practically meaningless difference is statistically significant, then you have wasted resources getting that much data. It would have been better to decide what a meaningful difference would be and designed the study to give you enough power to detect that difference but not smaller. And if we really want to split hairs, how do we define what parts of the lamb are on the right and which are on the left? If we define it by a line that by definition has equal number of hairs on each side then the answer to the above question becomes "Of Course it is".