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For example, for a fixed sample size, the probability of failing to reject a null hypothesis of equal population means decreases as the difference between population means increases.

A small *p*-value favors the alternative hypothesis. A small *p*-value means the observed data would not be very likely to occur if we believe the null hypothesis is true. So we believe in our data and disbelieve the null hypothesis. An easy (hopefully!) way to grasp this is to consider the situation where a professor states that you are just a 70% student. You doubt this statement and want to show that you are better that a 70% student. If you took a random sample of 10 of your previous exams and calculated the mean percentage of these 10 tests, which mean would be **less likely** to occur if in fact you were a 70% student (the null hypothesis): a sample mean of 72% or one of 90%? Obviously the 90% would be less likely and therefore would have a small probability (i.e. p-value).

Notice that the top part of the statistic is the difference between the sample mean and the null hypothesis. The bottom part of the calculation is the standard error of the mean.

**Definition: **Assuming that the null hypothesis is true, the p value isthe probability of obtaining a sample mean as extreme or more extreme than thesample mean actually obtained.

How small is "small?" Once we get the *p* value (probability) for an inferential statistic, we need to make a decision. Do we accept or reject the null hypothesis? What *p* value should we use as a cutoff?

**State the null hypothesis.** When you state the null hypothesis, you also have to state the alternate hypothesis. Sometimes it is easier to state the alternate hypothesis first, because that’s the researcher’s thoughts about the experiment. (opens in a new window).

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We have already seen how to do the first step, and have null and alternate hypotheses. The second step involves the calculation of the *t*-statistic for one mean, using the formula:

When testing hypotheses about a mean or mean difference, a *t*-distribution is used to find the p-value. This is a close cousin to the normal curve. T-Distributions are indexed by a quantity called degrees of freedom, calculated as df = n – 1 for the situation involving a test of one mean or test of mean difference.

The p value is just one piece of information you can use when deciding if your is true or not. You can use other values given by your test to help you decide. For example, if you run an, you’ll get a p value, an f-critical value and a .

In the above image, the results from the show a large p value (.244531, or 24.4531%), so you would not reject the null. However, there’s also another way you can decide: compare your f-value with your f-critical value. If the f-critical value is smaller than the f-value, you should reject the null hypothesis. In this particular test, the p value *and* the f-critical values are both very large so you do not have enough evidence to reject the null.

In an ideal world, you’ll have an alpha level. But if you do not, you can still use the following rough guidelines in deciding whether to support or reject the null hypothesis:

The null hypothesis is usually an hypothesis of "no difference" e.g. no difference between blood pressures in group A and group B. Define a null hypothesis for each study question clearly before the start of your study.

Basically, you reject the null hypothesis when your test value falls into the . There are four main ways you’ll compute test values and either support or reject your null hypothesis. Which method you choose depends mainly on if you have a proportion or a .

The null hypothesis can be thought of as a *nullifiable *hypothesis. That means you can nullify it, or reject it. What happens if you reject the null hypothesis? It gets replaced with the which is what you think might actually be true about a situation. For example, let’s say you think that a certain drug might be responsible for a spate of recent heart attacks. The drug company thinks the drug is safe. The null hypothesis is always the accepted hypothesis; in this example, the drug is on the market, people are using it, and it’s generally accepted to be safe. Therefore, the null hypothesis is that the drug is safe. The alternate hypothesis — the one you want to replace the null hypothesis, is that the drug *isn’t* safe. Rejecting the null hypothesis in this case means that you will have to prove that the drug is not safe.

If you have a , or are asked to find a p-value, follow these instructions to support or reject the null hypothesis. This method works if you are given an *and *if you are *not* given an alpha level. If you are given a , just subtract from 1 to get the alpha level. See: .

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