Understanding P-Value Confidence in Hypotheses Testing

[Moose352]

I think I'm confused with the concept of the P-Value. Take the hypothesis:

Ho: u = 5
Ha: u != 5

And then take another hypothesis:

Ho: u = 5
Ha: u > 5

I won't compute the p values, but it is easy to see that with the same statistic test, the p value for the first hypothesis will be twice that of the second, meaning that the statistic supports the second hypothesis much better than the first hypothesis. How can this be, especially when the second hypothesis is a sub case of the first hypothesis? What am I missing?

Thanks.

 

[Aaron Bex]

The p-value is a measure of how likely the data are to support the hypothesis. In this case, the p-value for the first hypothesis compares the likelihood of the observed data given that the true population mean is equal to 5, versus the likelihood of the observed data given that the true population mean is not equal to 5. The p-value for the second hypothesis compares the likelihood of the observed data given that the true population mean is greater than 5, versus the likelihood of the observed data given that the true population mean is not greater than 5.

The p-value for the first hypothesis will be smaller than the p-value for the second hypothesis because the second hypothesis is more restrictive than the first, meaning that it is less likely that the observed data would support it.

 

[tacman]

The concept of p-value can be confusing, so let's break it down. P-value is the probability of obtaining a test statistic at least as extreme as the one observed, assuming that the null hypothesis (Ho) is true. In simpler terms, it is the probability of getting the results we observed if the null hypothesis is actually true.

In your first hypothesis, the alternative hypothesis (Ha) is stating that the population mean (u) is not equal to 5. This means that the test is two-tailed, as there is no specific direction for the difference from 5. In this case, the p-value represents the probability of getting a test statistic equally or more extreme than the one observed in either direction (less than 5 or greater than 5).

On the other hand, in your second hypothesis, the alternative hypothesis (Ha) is stating that the population mean (u) is greater than 5. This is a one-tailed test, as there is only one direction for the difference from 5. Therefore, the p-value represents the probability of getting a test statistic as extreme or more extreme than the one observed in that specific direction (greater than 5).

Since the p-value in the second hypothesis only considers one direction (greater than 5), it will be smaller compared to the p-value in the first hypothesis, which considers both directions (less than 5 and greater than 5). This does not mean that the second hypothesis is a better fit for the data, but rather that the p-value is more specific in its interpretation.

In conclusion, the p-value represents the strength of evidence against the null hypothesis. A smaller p-value indicates stronger evidence against the null hypothesis, but it does not necessarily mean that the alternative hypothesis is a better fit for the data. It is important to consider the specific direction and interpretation of the p-value in relation to the alternative hypothesis. I hope this helps clarify the concept of p-value for you.