# Hypothesis Testing and Basic Concepts in Statistics

#### Basic Concepts of Hypothesis Testing in Statistics

In hypothesis testing, a population is a group of entities that share at least one thing in common. For example, all Americans who own a house are a population. A parameter of a population is an attribute that describes an entire population. For instance, the mean income of Americans who own a house is a parameter. The mean and standard deviation of a population are the two most common parameters. Because it is impossible to measure the parameters of a population, a random selection is taken out of the population. A random sample indicates that every member of a population has the same opportunity of being selected. After that, the parameters of the random selection are determined. The parameters of the population are then estimated to be close to those of the random sample.

For example, if we want to identify the mean income of the students of Westcliff University for the hypothesis testing, the population, in this case, is Westcliff’s students. The mean income is a parameter of the population. We then draw a random selection of 100 students and conclude that their mean income is $30,000/year (a statistic). We can say that the mean income of all Westcliff’s students is probably close to $30,000. In other words, the mean income of random selection enables us to make an educated guess about the mean income of the entire population. Generally, the bigger the selection is, the more accurate the result is.

#### Hypothesis Testing on One Population and Two Populations

Hypothesis testing refers to a statistical act that makes inferences about a population parameter based on the data of a sample taken from the population. The purpose of this test is to provide evidence regarding the plausibility of two hypotheses: the null hypothesis and the alternative hypothesis. The null hypothesis shows that there is no observed result in the experiment. We try to reject the null hypothesis. The alternative hypothesis is the opposite of the null hypothesis. We can determine if we should accept the alternative hypothesis based on the null hypothesis.

When testing the hypotheses on a population, we compare the population parameter with the value of the observations. Hypothetical examples can be testing whether the percentage of registered voters in Orange County is greater than 20% or testing whether Westcliff students graduate within five years, on average.

When testing the hypothesis on two populations, we compare the population parameters of two populations—for instance, testing whether students in Orange County perform better than students in LA County or testing whether the new hospital provides better service than the old hospital.

#### Possible Outcomes of Hypothesis Testing

Because the goal of doing hypothesis testing is to find evidence against the null hypothesis and reject it, there are two outcomes: to reject the null hypothesis or to retain it. If we reject the null hypothesis, we accept the alternative hypothesis. If we accept the null hypothesis, it means that there is not enough evidence to reject it.

If we consider errors, there are four possible outcomes:

- The null hypothesis is true, and the conclusion is to accept the null hypothesis
- The null hypothesis is true, but the conclusion is to reject the null hypothesis (type I error)
- The alternative hypothesis is true, but the conclusion is to reject the alternative hypothesis (type II error)
- The alternative hypothesis is true, and the conclusion is to accept the alternative hypothesis