# Hypothesis Testing

## The Two Types of Hypotheses in Hypothesis Testing

When trying to make a statement about a population parameter, we formulate a hypothesis that we can then put to the test by gathering a sample or conducting an experiment. However, before we can begin testing, we must establish two hypotheses: the null hypothesis and the alternative hypothesis.

### The Null Hypothesis and the Alternative Hypothesis

The null hypothesis, denoted as **H _{0}**, is the hypothesis that we assume to be true. On the other hand, the alternative hypothesis, denoted as

**H**, describes the potential outcome for the population parameter if the null hypothesis is incorrect.

_{A}The null hypothesis always states that the population parameter is equal to a specific value, while the alternative hypothesis differs depending on whether the test is one-tailed or two-tailed. In a one-tailed test, the null hypothesis will include either **≥** or **≤**, whereas in a two-tailed test, it will involve **≠**.

Additionally, we refer to the test statistic as the result of the experiment or sample from the population.

### How to Conduct a Hypothesis Test

The general method for conducting a hypothesis test involves assuming that the null hypothesis is true. Then, we calculate the probability of the test statistic occurring under the conditions of the null hypothesis. If this probability is less than the given significance level, denoted as **α**, we can reject the null hypothesis. If not, we do not have enough evidence to reject the null hypothesis. Note: it is sometimes said that we "accept" the null hypothesis, but this phrasing is not entirely accurate.

The significance level, **α**, is typically set at 5%. In some cases, we may also need to determine the critical region and values of a probability distribution.

The critical region is the area under the probability distribution where we would reject the null hypothesis if the test statistic falls within it. The critical value is the first value to fall within this region. In a one-tailed test, there will be one critical region and one critical value, while in a two-tailed test, there will be two of each as we must consider both ends of the distribution.

This can also be visualized on a graph, where the test statistic must fall within the critical region in order to reject the null hypothesis. In a one-tailed test, the critical region is represented by a specific area on the graph, while in a two-tailed test, there are two distinct critical regions, each representing a different end of the distribution.

### Using the Binomial Distribution for Hypothesis Testing

When dealing with a binomially distributed random variable **X**, written as **X ~ B(n,p)**, we are testing probabilities, meaning that the population parameter is **p**. This means that we must use **p** in defining our hypotheses.

#### One-Tailed Test

The steps for conducting a one-tailed test are as follows:

As an example, let's consider a standard six-sided die used in a board game that is thought to be biased because it does not roll a one as frequently as the other five values. In 40 rolls, one appears only three times. Using a 5% significance level, we can test whether this die is truly biased. Let **X** be the number of times a six-sided die was rolled, giving a result of one. Let **p** be the probability of obtaining a one in a roll of the die. By assuming that the null hypothesis is true, we can calculate the probability of observing a result of three or fewer ones out of 40 rolls, which turns out to be 0.028. This probability is less than the significance level of 0.05, allowing us to reject the null hypothesis and conclude that the die is indeed biased.

#### Two-Tailed Test

The steps for conducting a two-tailed test are nearly identical to those for a one-tailed test. However, in this case, we must divide the significance level in half and test for both ends of the distribution. For example:

A teacher believes that 30% of students watch football on a Saturday afternoon. The teacher asks 50 students, and 21 of them watch football on the weekend.

## Hypothesis Testing Simplified

Hypothesis testing is a statistical approach used to determine if a result is significant enough to reject the null hypothesis. In this process, we compare a sample to a population parameter to assess any differences. This article will break down the steps and concepts of hypothesis testing, specifically with the normal distribution, correlation, and key takeaways.

## The Normal Distribution and Hypothesis Testing

When conducting hypothesis testing with the normal distribution, we use the population mean (μ) as the parameter. We take a random sample (n) from the population, with a random variable (X) and a standard deviation (σ) that is normally distributed with μ. We then use the distribution of the sample mean to determine if the mean from the sample is statistically significant.

The process for hypothesis testing with the normal distribution is similar to that of the binomial distribution, but with a different population parameter. The critical region or value for the normal distribution can also be found by standardizing the test statistic or by using the inverse normal distribution on a calculator.

For example, a company claims that their bags of potatoes have a mean weight of 100kg, with a standard deviation of 4kg. An inspector takes a sample of 25 bags and finds the mean weight to be 106.4kg. At a 5% significance level, we test if there is enough evidence to reject the claim that the mean mass is greater than 100kg. Using the null hypothesis of μ=100, we find that there is insufficient evidence to reject it. Thus, there is no significant difference in the mean mass at the 5% level.

In a hypothesis test, a random sample (n) is taken from a normally distributed population with a mean (μ) and standard deviation (σ). The critical region for the test statistic can be found using the formula: θ = (x-μ)/(σ/√n).

## Correlation and Hypothesis Testing

Hypothesis testing for correlation focuses on determining if there is a significant correlation between two variables. The product moment correlation coefficient (PMCC) is used for this, with 1 representing a strong positive correlation, 0 representing no correlation, and -1 representing a strong negative correlation. The sample PMCC is denoted as r, while the population is denoted as ρ.

In a two-tailed test, we assess if the sample provides enough evidence to conclude that the population correlation is not zero, using r and ρ. In a one-tailed test, we determine if there is enough evidence to conclude that the population has either a positive or negative correlation, using r and ρ.

The critical region for r can be found using statistical tables and a formula booklet. For example, a teacher believes there is a correlation between shoe size and height and takes a sample of 50 students, finding a correlation of 0.34. Is there enough evidence to conclude a positive correlation at a 1% level of significance? The critical value from tables is r = 0.3281, and since 0.34> 0.3281, we can reject the null hypothesis and conclude that the sample suggests a positive correlation.

## The Key Takeaways of Hypothesis Testing

Hypothesis testing is used to determine if a result is significant enough to reject the null hypothesis. It involves comparing a sample to a population parameter and assessing any differences. The binomial hypothesis test evaluates the probability of events, while the normal hypothesis test evaluates the mean of a population. Lastly, the hypothesis test for correlation assesses the significance of a correlation with the population parameter.

## In Conclusion

Hypothesis testing is an essential tool in statistical analysis, allowing us to determine if a result is statistically significant. By understanding the concepts and steps involved, we can make informed decisions based on the data at hand and draw accurate conclusions.

## Understanding Hypothesis Testing: Definition and Process

Hypothesis testing is a crucial tool when it comes to analyzing data and drawing conclusions. It involves clearly defining the test statistic and population parameter, as well as stating the null and alternative hypotheses with a specified level of significance.

But how exactly is a hypothesis test conducted? The process starts by defining the test statistic and population parameter, which are numerical values that represent the data under analysis. Then, the null and alternative hypotheses are stated, representing the two possible outcomes of the test. The null hypothesis assumes that there is no significant difference or relationship between the variables being tested, while the alternative hypothesis proposes that there is a difference or relationship present.

Next, the probability of the observed value occurring under the null hypothesis is calculated and compared to the chosen significance level. This significance level is typically set at 5%. If the calculated probability is lower than the significance level, the null hypothesis can be rejected. On the other hand, if the probability is higher than the significance level, there is not enough evidence to reject the null hypothesis and it remains true.

Now, you may be wondering when different types of hypothesis tests should be used. A normal hypothesis test is appropriate when testing means, while a binomial hypothesis test is used for testing probabilities. For determining correlation, a correlation test, also known as the Pearson Product-Moment Correlation Coefficient (PMCC) test, is utilized.

But why is hypothesis testing important? The answer is simple: it helps us determine the statistical significance of our results. By comparing the observed data to the null hypothesis, we can determine if the results can be generalized to the larger population or if they are simply due to chance.