# Hypothesis Test for a Single Mean

A hypothesis test for a single mean is a research procedure by which a statistical inference is made through a decision about a single quantitative population parameter. This type of hypothesis test is a student t-distribution, where we analyze the more-accurate t-distribution of the sample, and calculating the test statistic “$t_{df}$“. A hypothesis test requires a single quantitative data set that will be analyzed, and a decision will be made regarding a claim about the population. Single mean hypothesis tests are a blunt tool to reject or fail to reject the null hypothesis, or original claim for the problem. It is good for broad decisions, but not to estimate the change in the sample. This website by Arkansas State University has a chapter from a statistics text book with good information regarding hypothesis tests.

The single mean hypothesis test is comprised of five steps: (1) Form a hypothesis, (2) Check conditions, (3) Calculate the test statistic, (4) Create the null distribution/ find the p-value, (5) Conclusion.

The test is as follows:

1. Form a Hypothesis:

In hypothesis testing, we form two hypotheses: the null hypothesis, ($H_{0}$), and the alternate hypothesis, ($H_{A}$).

$H_{0}$: The null hypothesis is the initial claim about the population based on prior evidence. It is assumed to be true, unless we can provide evidence beyond reasonable doubt that would cause us to reject the null hypothesis.Be careful: the null can only be set equal to a certain value. It cannot be greater than or less than a value.

$H_{A}$: The alternate hypothesis is the claim in opposition to the null hypothesis. It can be greater than, less than, or not equal to the value the null was set to.

Both the null and alternate hypothesis have to be stated in words, as well as symbolically. When the hypotheses are stated symbolically, $\mu$ is used because we are estimating the population.For example, if we want to test that 60% of eligible voters in Pakistan voted in the recent presidential election:

$H_{0}$ : $\mu$ = 0.60

In the recent Pakistani presidential election, 60% of eligible voters voted.

$H_{A}$: $\mu$ $\neq$ 0.60

In the recent Pakistani presidential election, there were not 60% of eligible voters who voted.

2. Check Conditions:

In order to proceed with any hypothesis test, conditions need to be checked to see if the hypothesis test in question can be done. This ensures we have a reliable analysis of data.For hypothesis tests for a single mean, there are three conditions that need to be checked:

(i). Randomness – The sample has to be random and representative of the population being studied.

(ii). Independence- The total sample needs to be less than than 10% of the population being studied. (e.g. 9 eligible voters out of 100)   Additionally, this means the probabilities of two events cannot be dependent on each other. A good example of this is when drawing cards out of a deck, the cards have to be put back into the deck, otherwise, the probability of drawing subsequent cards is changed.

(iii). Nearly normal sample distribution- For the real model to make sense, we need to know the population is normally distributed. This means the sample data is unimodal, symmetric, and there are no outliers. However, if the sample is larger than or equal to 40, the sample is large enough to yield a normal distribution.  Additionally, we need a Q-Q plot, which compares the sample data points to where they should be on a normal model.

3. Calculate the test statistic, tdf :

The test statistic gives a quantitative value to the behavior of the data. The University of North Carolina at Wilmington has good information on test statistics. For $t_{df}$,  “df” means degrees of freedom. The degrees of freedom in this case is n-1, where n is the number of events in the sample.

$t_{df}$ = $\frac{ \bar{y} - \mu}{ s/ \sqrt{n} }$

4. Find the null distribution and the p-value:

We use the test statistics and the degrees of freedom from the previous step to make a null distribution. There are infinitely many null distributions for the student t-test that are indexed by the degrees of freedom. The distributions are specific to the degrees of freedom, so this is an important part of the hypothesis test.

For example, if we had an alternate hypothesis that took a two-tailed, or $latex\neq$, counterpoint to the null hypothesis, and we found$t_{df}$ = 1.75,  the null distribution (also known as the t-distribution) will look like:

The tales are shaded to the left of the t-value of -1.75, and to the right of the positive t-value of +1.75.
The shaded regions give us the p-value. The p-value is the probability, assuming the null hypothesis is true, of getting a sample that is as extreme or more extreme than that of the present sample
completely by chance.
5. Conclusion:
The conclusion to a hypothesis test is determined by comparing the p-value found in the previous step to a significance level. The significance level is the probability that the null hypothesis is rejected even though it is true, and the probability that the researcher is wrong.  It is the cutoff value for P that is chosen before the test is run, and it is the framework by which the null hypothesis is rejected, or if the researcher fails to reject the null. A common mistake that should never be made is to accept the null.
Here are the cases for a conclusion to hypothesis testing:
P <α : Reject the null hypothesis
If the null is rejected, we conclude by using the following sentence as the template:

“There is sufficient evidence that $H_{A}$ is true”

P > α: Fail to reject the null hypothesis

If we fail to reject the null, we conclude by using the following sentence as the template:

“There is insufficient evidence that $H_{A}$ is true”

The conclusions should always be contextually meaningful. For another simple explanation of hypothesis testing, visit the University College Dublin Statistics Page. Another good website for single mean hypothesis testing is the Stat Trek website that provides explanations, as well as some good examples.