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Null and Alternative Hypotheses for a Mean

The alternative hypothesis might also be that the new drug is better, on average, than the current drug.

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Support or Reject Null Hypothesis

In many statistical tests, you’ll want to either reject or support the . For elementary statistics students, the term can be a tricky term to grasp, partly because the name “null hypothesis” doesn’t make it clear about what the null hypothesis actually is!

Click the link the skip to the situation you need to support or reject null hypothesis for:

The significance level (also known as the "critical value" or "alpha") you should use depends on the costs of different kinds of errors. With a significance level of 0.05, you have a 5% chance of rejecting the null hypothesis, even if it is true. If you try 100 different treatments on your chickens, and none of them really change the sex ratio, 5% of your experiments will give you data that are significantly different from a 1:1 sex ratio, just by chance. In other words, 5% of your experiments will give you a false positive. If you use a higher significance level than the conventional 0.05, such as 0.10, you will increase your chance of a false positive to 0.10 (therefore increasing your chance of an embarrassingly wrong conclusion), but you will also decrease your chance of a false negative (increasing your chance of detecting a subtle effect). If you use a lower significance level than the conventional 0.05, such as 0.01, you decrease your chance of an embarrassing false positive, but you also make it less likely that you'll detect a real deviation from the null hypothesis if there is one.

Null hypothesis: μ = 72 Alternative hypothesis: μ ≠72

Null hypothesis: μ = 72  Alternative hypothesis: μ ≠72

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 .

You should decide whether to use the one-tailed or two-tailed probability before you collect your data, of course. A one-tailed probability is more powerful, in the sense of having a lower chance of false negatives, but you should only use a one-tailed 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 one-tailed 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 sex-determination 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 two-tailed probability. In addition, people are skeptical of one-tailed probabilities, especially if a one-tailed probability is significant and a two-tailed probability would not be significant (as in our chocolate-eating chicken example). Unless you provide a very convincing explanation, people may think you decided to use the one-tailed probability after you saw that the two-tailed probability wasn't quite significant, which would be cheating. It may be easier to always use two-tailed probabilities. For this handbook, I will always use two-tailed probabilities, unless I make it very clear that only one direction of deviation from the null hypothesis would be interesting.

the null hypothesis is rejected when it is true b.

Broken down into English, that’s H0 (The null hypothesis): μ (the average) = (is equal to) 8.2

Because this is a two-sided alternative hypothesis, the p-value is the combined area to the right of 2.47 and the left of −2.47 in a t-distribution with 35 – 1 = 34 degrees of freedom.

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.

In English again, that’s H1 (The alternate hypothesis): μ (the average) ≠ (is not equal to) 8.2
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  • One can never prove the truth of a statistical (null) hypothesis.

    A one-sided hypothesis claims that a parameter is either larger smaller than thevalue given by the null hypothesis.

  • failing to reject the null hypothesis when it is false.

    Once null and alternative hypotheses have been formulated for a particular claim, the next stepis to compute a .

  • failing to reject the null hypothesis when it is true.

    If you are able to reject the null hypothesis in Step 2, you can replace it with the alternate hypothesis.

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rejecting the null hypothesis when it is true.

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 low-cost 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.

rejecting the null hypothesis when it is false.

A Bayesian would insist that you put in numbers just how likely you think the null hypothesis and various values of the alternative hypothesis are, before you do the experiment, and I'm not sure how that is supposed to work in practice for most experimental biology. But the general concept is a valuable one: as Carl Sagan summarized it, "Extraordinary claims require extraordinary evidence."

rejecting the null hypothesis when the alternative is true.

Now instead of testing 1000 plant extracts, imagine that you are testing just one. If you are testing it to see if it kills beetle larvae, you know (based on everything you know about plant and beetle biology) there's a pretty good chance it will work, so you can be pretty sure that a P value less than 0.05 is a true positive. But if you are testing that one plant extract to see if it grows hair, which you know is very unlikely (based on everything you know about plants and hair), a P value less than 0.05 is almost certainly a false positive. In other words, if you expect that the null hypothesis is probably true, a statistically significant result is probably a false positive. This is sad; the most exciting, amazing, unexpected results in your experiments are probably just your data trying to make you jump to ridiculous conclusions. You should require a much lower P value to reject a null hypothesis that you think is probably true.

not rejecting the null hypothesis when the alternative is true.

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.

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