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gung - Reinstate Monica
gung - Reinstate Monica
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I don't see any reason why having $\mu=25$ is impossible. After all, $\mu$ has to equal some number. It is common in statistical circles to hear people say that the null hypothesis, $\text H_0: \mu_{\text{treatment}}=\mu_{\text{control}}$, must be false (which is a statement that I don't necessarily agree with, but that's another discussion). Your situation, however, is (superficially similar, but) different from this.

Another issue that might be confusing you is that you refer to "the probability of μ equaling 25". This is a very common misconception of classical (frequentistFrequentist) hypothesis testing. Properly speaking, there is no 'probability' of $\mu$ equaling 25, or any other number. You could say that the probability is either $1$ or $0$, but you don't know which. Nonetheless, $\mu$ isn't a random variable; it's a parameter, that is, an unknown constant. The only probability distribution that can be assigned to this is the degenerate distributiondegenerate distribution.

 

So, what are you doing?So, what are you doing? When you perform a hypothesis test like what I suspect you are doingperforming, you checkare checking to see what is the probability of getting a value (specifically a sample mean) as extreme or more extreme than yours, if the null hypothesis is true. The answer to that question is your p-value. After having done this, there are at least two valid ways of interpreting that p-value. First

First, Fisher thought you should interpret it as a continuous measure of evidence against the null hypothesis. (Note that there is no alternative hypothesis or 'bright line' here, .04 is about the same as .06.) Another

Another possibility is to use the p-value as part of a decision making process, in keeping with the ideas of Neyman and Pearson. From this perspective, you are trying to differentiate between two hypotheses, and you designed your study explicitly to do so. For example, you conducted a power analysispower analysis, and set $\alpha$ and $\beta$ to levels of type I and type II errortype I and type II error rates that you are willing to live with in the long run. After having analyzed your data and made a decision, you will either have made one of those errors or a correct decision, but you will never know which; you will only know that over all the times you engaged in this activity, you will have made a certain percentage of type I (type II) errors for those cases when the null hypothesis was actually true (false).

I don't see any reason why having $\mu=25$ is impossible. After all, $\mu$ has to equal some number. It is common in statistical circles to hear people say that the null hypothesis, $\text H_0: \mu_{\text{treatment}}=\mu_{\text{control}}$, must be false (which is a statement that I don't necessarily agree with, but that's another discussion). Your situation, however, is (superficially similar, but) different from this.

Another issue that might be confusing you is that you refer to "the probability of μ equaling 25". This is a very common misconception of classical (frequentist) hypothesis testing. Properly speaking, there is no 'probability' of $\mu$ equaling 25, or any other number. You could say that the probability is either $1$ or $0$, but you don't know which. Nonetheless, $\mu$ isn't a random variable; it's a parameter, that is, an unknown constant. The only probability distribution that can be assigned to this is the degenerate distribution.

So, what are you doing? When you perform a hypothesis test like what I suspect you are doing, you check to see what is the probability of getting a value (specifically a sample mean) as extreme or more extreme than yours, if the null hypothesis is true. The answer to that question is your p-value. After having done this, there are at least two valid ways of interpreting that p-value. First, Fisher thought you should interpret it as a continuous measure of evidence against the null hypothesis. (Note that there is no alternative hypothesis or 'bright line' here, .04 is about the same as .06.) Another possibility is to use the p-value as part of a decision making process, in keeping with the ideas of Neyman and Pearson. From this perspective, you are trying to differentiate between two hypotheses, and you designed your study to do so. For example, you conducted a power analysis, and set $\alpha$ and $\beta$ to levels of type I and type II error rates that you are willing to live with in the long run. After having analyzed your data and made a decision, you will either have made one of those errors or a correct decision, but you will never know which; you will only know that over all the times you engaged in this activity, you will have made a certain percentage of type I (type II) errors for those cases when the null hypothesis was actually true (false).

I don't see any reason why having $\mu=25$ is impossible. After all, $\mu$ has to equal some number. It is common in statistical circles to hear people say that the null hypothesis, $\text H_0: \mu_{\text{treatment}}=\mu_{\text{control}}$, must be false (which is a statement that I don't necessarily agree with, but that's another discussion). Your situation, however, is (superficially similar, but) different from this.

Another issue that might be confusing you is that you refer to "the probability of μ equaling 25". This is a very common misconception of classical (Frequentist) hypothesis testing. Properly speaking, there is no 'probability' of $\mu$ equaling 25, or any other number. You could say that the probability is either $1$ or $0$, but you don't know which. Nonetheless, $\mu$ isn't a random variable; it's a parameter, that is, an unknown constant. The only probability distribution that can be assigned to this is the degenerate distribution.

 

So, what are you doing? When you perform a hypothesis test like what I suspect you are performing, you are checking to see what is the probability of getting a value (specifically a sample mean) as extreme or more extreme than yours, if the null hypothesis is true. The answer to that question is your p-value. After having done this, there are at least two valid ways of interpreting that p-value.

First, Fisher thought you should interpret it as a continuous measure of evidence against the null hypothesis. (Note that there is no alternative hypothesis or 'bright line' here, .04 is about the same as .06.)

Another possibility is to use the p-value as part of a decision making process, in keeping with the ideas of Neyman and Pearson. From this perspective, you are trying to differentiate between two hypotheses, and you designed your study explicitly to do so. For example, you conducted a power analysis, and set $\alpha$ and $\beta$ to levels of type I and type II error rates that you are willing to live with in the long run. After having analyzed your data and made a decision, you will either have made one of those errors or a correct decision, but you will never know which; you will only know that over all the times you engaged in this activity, you will have made a certain percentage of type I (type II) errors for those cases when the null hypothesis was actually true (false).

Source Link
gung - Reinstate Monica
gung - Reinstate Monica
  • 147.5k
  • 89
  • 406
  • 717

I don't see any reason why having $\mu=25$ is impossible. After all, $\mu$ has to equal some number. It is common in statistical circles to hear people say that the null hypothesis, $\text H_0: \mu_{\text{treatment}}=\mu_{\text{control}}$, must be false (which is a statement that I don't necessarily agree with, but that's another discussion). Your situation, however, is (superficially similar, but) different from this.

Another issue that might be confusing you is that you refer to "the probability of μ equaling 25". This is a very common misconception of classical (frequentist) hypothesis testing. Properly speaking, there is no 'probability' of $\mu$ equaling 25, or any other number. You could say that the probability is either $1$ or $0$, but you don't know which. Nonetheless, $\mu$ isn't a random variable; it's a parameter, that is, an unknown constant. The only probability distribution that can be assigned to this is the degenerate distribution.

So, what are you doing? When you perform a hypothesis test like what I suspect you are doing, you check to see what is the probability of getting a value (specifically a sample mean) as extreme or more extreme than yours, if the null hypothesis is true. The answer to that question is your p-value. After having done this, there are at least two valid ways of interpreting that p-value. First, Fisher thought you should interpret it as a continuous measure of evidence against the null hypothesis. (Note that there is no alternative hypothesis or 'bright line' here, .04 is about the same as .06.) Another possibility is to use the p-value as part of a decision making process, in keeping with the ideas of Neyman and Pearson. From this perspective, you are trying to differentiate between two hypotheses, and you designed your study to do so. For example, you conducted a power analysis, and set $\alpha$ and $\beta$ to levels of type I and type II error rates that you are willing to live with in the long run. After having analyzed your data and made a decision, you will either have made one of those errors or a correct decision, but you will never know which; you will only know that over all the times you engaged in this activity, you will have made a certain percentage of type I (type II) errors for those cases when the null hypothesis was actually true (false).