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How to Determine a p-Value When Testing a Null Hypothesis

Hypothesis testing is very important in the scientific community and is necessary for advancing theories and ideas. Statistical hypothesis tests are not just designed to select the more likely of two hypotheses—a test will remain with the null hypothesis until there's enough evidence to support the alternative hypothesis. Now you have seen several examples of hypothesis testing and you can better understand why it is so important. For more information on types of hypotheses see .

All null hypotheses include an equal sign in them.

When you set up a hypothesis test to determine the validity of a statistical claim, you need to define both a null hypothesis and an alternative hypothesis.

the null hypothesis is rejected when it is true b.

The p-value is p = 0.236. This is not below the .05 standard, so we do not reject the null hypothesis. Thus it is possible that the true value of the population mean is 72. The 95% confidence interval suggests the mean could be anywhere between 67.78 and 73.06.

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.

the null hypothesis is not rejected when it is false c.

The p-value is p = 0.019. This is below the .05 standard, so the result is statistically significant. This means we decide in favor of the alternative hypothesis. We're deciding that the population mean is not 72.

Every hypothesis test contains a set of two opposing statements, or hypotheses, about a population parameter. The first hypothesis is called the denoted H0. The null hypothesis always states that the population parameter is to the claimed value. For example, if the claim is that the average time to make a name-brand ready-mix pie is five minutes, the statistical shorthand notation for the null hypothesis in this case would be as follows:

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failing to reject the null hypothesis when it is false.

So what is needed is not just a system of null hypothesis testing but also a system for telling us precisely how large the effects we see in our data really are.

failing to reject the null hypothesis when it is true.

When you about a , you can use your test statistic to decide whether to reject the null hypothesis, H0. You make this decision by coming up with a number, called a -value.

rejecting the null hypothesis when it is true.

In general, a p-value is the probability that the test statistic would "lean" as much (or more) toward the alternative hypothesis as it does if the real truth is the null hypothesis.

rejecting the null hypothesis when it is false.

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.

rejecting the null hypothesis when the alternative is true.

where the observed sample mean difference, μ0 = value specified in null hypothesis, sd = standard deviation of the differences in the sample measurements and n = sample size. For instance, if we wanted to test for a difference in mean SAT Math and mean SAT Verbal scores, we would random sample subjects, record their SATM and SATV scores in two separate columns, then create a third column that contained the differences between these scores. Then the sample mean and sample standard deviation would be those that were calculated on this column of differences.

not rejecting the null hypothesis when the alternative is true.

where the observed sample mean, μ0 = value specified in null hypothesis, s = standard deviation of the sample measurements and n = the number of differences.

the null hypothesis is rejected when it is true.

Before actually conducting a hypothesis test, you have to put two possible hypotheses on the table — the null hypothesis is one of them. But, if the null hypothesis is rejected (that is, there was sufficient evidence against it), what’s your alternative going to be? Actually, three possibilities exist for the second (or alternative) hypothesis, denoted Ha. Here they are, along with their shorthand notations in the context of the pie example:

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