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Support or Reject Null Hypothesis
Here are three experiments to illustrate when the different approaches to statistics are appropriate. In the first experiment, you are testing a plant extract on rabbits to see if it will lower their blood pressure. You already know that the plant extract is a diuretic (makes the rabbits pee more) and you already know that diuretics tend to lower blood pressure, so you think there's a good chance it will work. If it does work, you'll do more lowcost animal tests on it before you do expensive, potentially risky human trials. Your prior expectation is that the null hypothesis (that the plant extract has no effect) has a good chance of being false, and the cost of a false positive is fairly low. So you should do frequentist hypothesis testing, with a significance level of 0.05.
Another alternative to frequentist statistics is Bayesian statistics. A key difference is that Bayesian statistics requires specifying your best guess of the probability of each possible value of the parameter to be estimated, before the experiment is done. This is known as the "prior probability." So for your chickensex experiment, you're trying to estimate the "true" proportion of male chickens that would be born, if you had an infinite number of chickens. You would have to specify how likely you thought it was that the true proportion of male chickens was 50%, or 51%, or 52%, or 47.3%, etc. You would then look at the results of your experiment and use the information to calculate new probabilities that the true proportion of male chickens was 50%, or 51%, or 52%, or 47.3%, etc. (the posterior distribution).
which is equivalent to rejecting the null hypothesis:
You should decide whether to use the onetailed or twotailed probability before you collect your data, of course. A onetailed probability is more powerful, in the sense of having a lower chance of false negatives, but you should only use a onetailed probability if you really, truly have a firm prediction about which direction of deviation you would consider interesting. In the chicken example, you might be tempted to use a onetailed probability, because you're only looking for treatments that decrease the proportion of worthless male chickens. But if you accidentally found a treatment that produced 87% male chickens, would you really publish the result as "The treatment did not cause a significant decrease in the proportion of male chickens"? I hope not. You'd realize that this unexpected result, even though it wasn't what you and your farmer friends wanted, would be very interesting to other people; by leading to discoveries about the fundamental biology of sexdetermination in chickens, in might even help you produce more female chickens someday. Any time a deviation in either direction would be interesting, you should use the twotailed probability. In addition, people are skeptical of onetailed probabilities, especially if a onetailed probability is significant and a twotailed probability would not be significant (as in our chocolateeating chicken example). Unless you provide a very convincing explanation, people may think you decided to use the onetailed probability after you saw that the twotailed probability wasn't quite significant, which would be cheating. It may be easier to always use twotailed probabilities. For this handbook, I will always use twotailed probabilities, unless I make it very clear that only one direction of deviation from the null hypothesis would be interesting.
The probability that was calculated above, 0.030, is the probability of getting 17 or fewer males out of 48. It would be significant, using the conventional PP=0.03 value found by adding the probabilities of getting 17 or fewer males. This is called a onetailed probability, because you are adding the probabilities in only one tail of the distribution shown in the figure. However, if your null hypothesis is "The proportion of males is 0.5", then your alternative hypothesis is "The proportion of males is different from 0.5." In that case, you should add the probability of getting 17 or fewer females to the probability of getting 17 or fewer males. This is called a twotailed probability. If you do that with the chicken result, you get P=0.06, which is not quite significant.
which is equivalent to rejecting the null hypothesis:
This number, 0.030, is the P value. It is defined as the probability of getting the observed result, or a more extreme result, if the null hypothesis is true. So "P=0.030" is a shorthand way of saying "The probability of getting 17 or fewer male chickens out of 48 total chickens, IF the null hypothesis is true that 50% of chickens are male, is 0.030."
After you do a statistical test, you are either going to reject or accept the null hypothesis. Rejecting the null hypothesis means that you conclude that the null hypothesis is not true; in our chicken sex example, you would conclude that the true proportion of male chicks, if you gave chocolate to an infinite number of chicken mothers, would be less than 50%.
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Probability of incorrectly rejecting a true null hypothesis
Let's close this example by formalizing the definition of a Pvalue, as well as summarizing the Pvalue approach to conducting a hypothesis test.
3.2  Hypothesis Testing (Pvalue approach)  Statistics
Because the Pvalue 0.055 is (just barely) greater than the significance level α = 0.05, we barely fail to reject the null hypothesis. Again, we would say that there is insufficient evidence at the α = 0.05 level to conclude that the sample proportion differs significantly from 0.90.
Probability that Null Hypothesis is True  Cross Validated
Often, the people who claim to avoid hypothesis testing will say something like "the 95% confidence interval of 25.9 to 47.4% does not include 50%, so we conclude that the plant extract significantly changed the sex ratio." This is a clumsy and roundabout form of hypothesis testing, and they might as well admit it and report the P value.
Probability that Null Hypothesis is True
Alternatively (and the way I prefer to think of Pvalues), the Pvalue is the probability that we'd observe a more extreme statistic than we did if the null hypothesis were true.
probability that, if the null hypothesis ..
A related criticism is that a significant rejection of a null hypothesis might not be biologically meaningful, if the difference is too small to matter. For example, in the chickensex experiment, having a treatment that produced 49.9% male chicks might be significantly different from 50%, but it wouldn't be enough to make farmers want to buy your treatment. These critics say you should estimate the effect size and put a on it, not estimate a P value. So the goal of your chickensex experiment should not be to say "Chocolate gives a proportion of males that is significantly less than 50% (P=0.015)" but to say "Chocolate produced 36.1% males with a 95% confidence interval of 25.9 to 47.4%." For the chickenfeet experiment, you would say something like "The difference between males and females in mean foot size is 2.45 mm, with a confidence interval on the difference of ±1.98 mm."
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