What is a *population parameter* in the context of a hypothesis testing question? When you are working on a homework assignment, you will likely be asked at some point to establish if something is a statistic or a parameter. In that context, a parameter is referring to a population parameter.

The idea of population parameter also arises when talking about hypotheses, namely, the null and alternative hypotheses. These hypotheses are defined as a *statement about a population parameter*. In order words, you are making a claim about the numerical value of a population parameter.

### Sample Information versus Population Information

Let us start setting the record straight: A population parameter is simply a number that determines the probabilistic behavior of a distribution, potentially together with other parameters. That is it. Population parameters are simply numerical values that determine of probability distribution. And this can be said in a more statistical context. Indeed, a population parameter is a (fixed, non-random) numerical value that determines the probabilistic behavior of a population being studied.

Consider the following example: You are the manager at a factory of electronic parts and components, and are interested in studying the mean duration of a specific electronic component called the *M23*. The population for this study is the set of all possible durations of the M23. The duration of the M23 is random in nature (it is not always the same, it always varies), and the engineers know it has an exponential distribution.

The engineers, by assuming the distribution of durations is exponential, they know that there exist a number \(\beta\) that determines the distribution. Indeed, once you fix the value of \(\beta\), the distribution (density) of durations of the M23 component takes the form:

\[f\left( x \right)=\frac{1}{\beta }{{e}^{-\frac{x}{\beta }}}\]

It turns out that this parameter \(\beta\) is *the population parameter of interest* for the engineers at this company.

Observe that furthermore, using a bit of Calculus, and using the definition of population mean, the population mean for this distribution is

\[\int\limits_{-\infty }^{\infty }{f\left( x \right)dx}=\int\limits_{-\infty }^{\infty }{\frac{1}{\beta }{{e}^{-\frac{x}{\beta }}}dx}=\beta \]

So it turns out that in this case, the population parameter of interest is the population mean, but it does not have to be like that every time.

KEEP THIS IN MIND: A POPULATION PARAMETER IS A NUMBER THAT DETERMINES A PROBABILITY DISTRIBUTION

In other, it is number that once you plug it in to the expression for the probability distribution allows you to have a function that can be evaluated for some range of x values. IT DOES NOT HAVE TO BE NECESSARILY the population mean or the population variance, but often time it is.

### Distinguishing Between a Statistic and a Parameter

Finally, a word of practical advice: how do you distinguish between a statistic and a parameter? That is a question frequently asked in Stats tests and homework. This is how you do it: You need to ask yourself, is the quantity you are being asked about calculated using sample information? If the answer to that question is yes, then you have a statistics. If not, then you likely have a parameter.

For example, when a questions reads like "a sample of 15 people is collected and you compute the mean height of those 15 people, is that a statistic or a parameter?" Then you need to ask yourself how you compute that quantity, and what you do is take all the values from the sample and take the arithmetic mean of those 15 values. So then, you ARE using sample information, and therefore, you have a statistic instead of a parameter.

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