The second type of inferential statistics, hypothesis testing (also called a *test of significance*), is very important in scientific fields. Basically, you use a hypothesis test when you want to investigate statements (or beliefs) about a characteristic of one or more populations. In this sub-competency you are introduced to the major ideas, computations, and conclusions of hypothesis testing. You will also see the types of errors (or mistakes) that can be made when using hypothesis tests.

An example will illustrate the language and all necessary components of a hypothesis test.

**Example: **What Is the “Normal” Human Body Temperature?

We all believe that the acceptable, or normal, human body temperature is **98.6°F**. How do we know this? Have any of us done our own testing or research to prove this? No! Our parents or friends or doctors told us this. So again how do we *know* that the normal human body temperature is 98.6°F? The answer is that we *don’t* know for sure…we take it on faith that 98.6°F is correct. This belief, in statistical terms, is called the *null hypothesis*, and is denoted by ** H_{0}**. A null hypothesis is an underlying belief about a characteristic of a population.

To test this deeply ingrained belief, i.e., to test the validity of the null hypothesis *H*_{0}, doctors from the University of Maryland took a sample of *n* = 106 body temperatures from healthy subjects. Their sample results were *x̄* = 98.2°F and *s* = 0.62°F.

The question to answer now is this:

Do the results from the sample support the null hypothesis or do the results lead us to believe that the normal human body temperature is different from 98.6°F?

The two numbers, 98.6°F and 98.2°F, are extremely close mathematically…they differ by only 0.4°F. But you have to keep the main issue of statistics in mind: *data varies*. In terms of the variation of the sample results, *s* = 0.62°F and the size of our sample, *n* = 106, how close are the two values 98.6°F and 98.2°F?

This is where the *z*-score will come into play again. We need to determine how different these two values are *in terms of the variation and size of the sample*. Recall that the sample mean has a distribution that is normal with mean *μ* and standard deviation *s*/√*n*. Performing the calculation with this in mind gives us our *test statistic*:

This means that the two numbers 98.6°F and 98.2°F actually differ by more than 6 standard deviations! Recalling the empirical rule discussed in subcompetency 3, roughly 99.7% of all data falls within 3 standard deviation of the mean. So, if the true population mean human body temperature really is 98.6°F, then our sample produced a result that differs from the true population mean by more than 6 standard deviations. What are the chances this would happen? In other words, how likely is it that such a large sample (*n* = 106) will produce a result that is so different from what is believed to be the true population mean? We’ll learn how to determine the exact probability, or likelihood, of this happening in the next subcompetency.

Finally, what are the conclusions we can draw from our test of the null hypothesis? There are only two choices:

- The true population mean really is 98.6°F and we were just really unlucky to get a sample result that was so different from the true population mean. It turns out, we’ll find later, that the probability of getting a sample result this different from the claimed mean is:

0.00000000142

In other words, we would only expect to get a sample this different in (roughly) 14 out of ever 10 **billion** samples of size *n* = 106. Were we just that unlucky? Or…

- The true population mean is not 98.6°F. In fact, since our sample result of
*x̄*= 98.2°F is so far below 98.6°F, our data is suggesting that the true population mean is likely lower than 98.6°F. We cannot claim that the true population mean is the same as the sample mean,*μ*=*x̄***= 98.2°F (because data varies!!)**, but our sample does suggest that a normal human body temperature is likely much lower than what is commonly believed.

Our best conclusion statistically would be no. 2, that the true population mean is not 98.6°F. Therefore, our data and results are strongly suggesting that we need to **reject** the belief that the normal human body temperature is 98.6°F. Statistically we say that, “we reject the null hypothesis *H*_{0} because our sample data provides strong evidence that the temperature is different (most likely a bit lower) than 98.2°F.”