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In the olden days, when people looked up P values in printed tables, they would report the results of a statistical test as "PPP>0.10", etc. Nowadays, almost all computer statistics programs give the exact P value resulting from a statistical test, such as P=0.029, and that's what you should report in your publications. You will conclude that the results are either significant or they're not significant; they either reject the null hypothesis (if P is below your pre-determined significance level) or don't reject the null hypothesis (if P is above your significance level). But other people will want to know if your results are "strongly" significant (P much less than 0.05), which will give them more confidence in your results than if they were "barely" significant (P=0.043, for example). In addition, other researchers will need the exact P value if they want to combine your results with others into a .
It is important to distinguish between biological null and alternative hypotheses and statistical null and alternative hypotheses. "Sexual selection by females has caused male chickens to evolve bigger feet than females" is a biological alternative hypothesis; it says something about biological processes, in this case sexual selection. "Male chickens have a different average foot size than females" is a statistical alternative hypothesis; it says something about the numbers, but nothing about what caused those numbers to be different. The biological null and alternative hypotheses are the first that you should think of, as they describe something interesting about biology; they are two possible answers to the biological question you are interested in ("What affects foot size in chickens?"). The statistical null and alternative hypotheses are statements about the data that should follow from the biological hypotheses: if sexual selection favors bigger feet in male chickens (a biological hypothesis), then the average foot size in male chickens should be larger than the average in females (a statistical hypothesis). If you reject the statistical null hypothesis, you then have to decide whether that's enough evidence that you can reject your biological null hypothesis. For example, if you don't find a significant difference in foot size between male and female chickens, you could conclude "There is no significant evidence that sexual selection has caused male chickens to have bigger feet." If you do find a statistically significant difference in foot size, that might not be enough for you to conclude that sexual selection caused the bigger feet; it might be that males eat more, or that the bigger feet are a developmental byproduct of the roosters' combs, or that males run around more and the exercise makes their feet bigger. When there are multiple biological interpretations of a statistical result, you need to think of additional experiments to test the different possibilities.
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Since the biologist's test statistic, t* = -4.60, is less than -1.6939, the biologist rejects the null hypothesis. That is, the test statistic falls in the "critical region." There is sufficient evidence, at the α = 0.05 level, to conclude that the mean height of all such sunflower seedlings is less than 15.7 cm.
If the biologist set her significance level α at 0.05 and used the critical value approach to conduct her hypothesis test, she would reject the null hypothesis if her test statistic t* were less than -1.6939 (determined using statistical software or a t-table):
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Usually, the null hypothesis is boring and the alternative hypothesis is interesting. For example, let's say you feed chocolate to a bunch of chickens, then look at the sex ratio in their offspring. If you get more females than males, it would be a tremendously exciting discovery: it would be a fundamental discovery about the mechanism of sex determination, female chickens are more valuable than male chickens in egg-laying breeds, and you'd be able to publish your result in Science or Nature. Lots of people have spent a lot of time and money trying to change the sex ratio in chickens, and if you're successful, you'll be rich and famous. But if the chocolate doesn't change the sex ratio, it would be an extremely boring result, and you'd have a hard time getting it published in the Eastern Delaware Journal of Chickenology. It's therefore tempting to look for patterns in your data that support the exciting alternative hypothesis. For example, you might look at 48 offspring of chocolate-fed chickens and see 31 females and only 17 males. This looks promising, but before you get all happy and start buying formal wear for the Nobel Prize ceremony, you need to ask "What's the probability of getting a deviation from the null expectation that large, just by chance, if the boring null hypothesis is really true?" Only when that probability is low can you reject the null hypothesis. The goal of statistical hypothesis testing is to estimate the probability of getting your observed results under the null hypothesis.
There are different ways of doing statistics. The technique used by the vast majority of biologists, and the technique that most of this handbook describes, is sometimes called "frequentist" or "classical" statistics. It involves testing a null hypothesis by comparing the data you observe in your experiment with the predictions of a null hypothesis. You estimate what the probability would be of obtaining the observed results, or something more extreme, if the null hypothesis were true. If this estimated probability (the P value) is small enough (below the significance value), then you conclude that it is unlikely that the null hypothesis is true; you reject the null hypothesis and accept an alternative hypothesis.
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Since the quality control specialist's test statistic, t* = 1.54, is not less than -2.2622 nor greater than 2.2622, the qualtiy control specialist fails to reject the null hypothesis. That is, the test statistic does not fall in the "critical region." There is insufficient evidence, at the α = 0.05 level, to conclude that the mean thickness of all of the manufacturer's spearmint gum differs from 7.5 one-hundredths of an inch.
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If the quality control specialist sets his significance level α at 0.05 and used the critical value approach to conduct his hypothesis test, he would reject the null hypothesis if his test statistic t* were less than -2.2622 or greater than 2.2622 (determined using statistical software or a t-table):
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Since the engineer's test statistic, t* = 1.22, is not greater than 1.7109, the engineer fails to reject the null hypothesis. That is, the test statistic does not fall in the "critical region." There is insufficient evidence, at the α = 0.05 level, to conclude that the mean Brinell hardness of all such ductile iron pieces is greater than 170.
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If the engineer set his significance level α at 0.05 and used the critical value approach to conduct his hypothesis test, he would reject the null hypothesis if his test statistic t* were greater than 1.7109 (determined using statistical software or a t-table):
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Hypothesis testing is not set up so that you can absolutely prove a null hypothesis. Therefore, when you do not find evidence against the null hypothesis, you fail to reject the null hypothesis. When you do find strong enough evidence against the null hypothesis, you reject the null hypothesis. When presenting the results of a hypothesis test, include the descriptive statistics in your conclusions as well. Report exact p-values rather than a certain range. For example, "The intubation rate differed significantly by patient age with younger patients have a lower rate of successful intubation (p=0.02)." Here are two more examples with the conclusion stated in several different ways.
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