Hypothesis testing is a method used to determine whether an assumption or claim about a population is true or false. There are two main types of hypothesis testing procedures: parametric and non-parametric.

Parametric tests assume that the variables being studied are measured on an interval scale, meaning that the data is continuous and has a meaningful zero point. Examples of parametric tests include Z-test and T-test.

Z-test is used when the sample size is large (typically greater than 30) and the population standard deviation is known. It is used to determine whether there is a significant difference between the sample mean and population mean.

T-test, on the other hand, is used when the sample size is small (typically less than 30) and the population standard deviation is unknown. It is also used to determine whether there is a significant difference between the sample mean and population mean.

Non-parametric tests, on the other hand, assume that the variables being studied are measured on an ordinal scale, meaning that the data is categorical and has no meaningful zero point. Examples of non-parametric tests include chi-squared test and Wilcoxon rank-sum test.

**Hypothesis Testing**

Hypothesis Testing is a branch of Inferential Statistics used to statistically test the truth of a statement and draw conclusions about whether to accept or reject the idea.

Brief statements or assumptions made to be tested for validity are called hypotheses or hypotheses.

The purpose of Hypothesis Testing is to establish a basis so that it can collect evidence in the form of data in determining whether to reject or accept the truth of the statements or assumptions made.

Hypothesis testing can also provide confidence in objective decision-making.

Examples of Hypothesis Statements that must be tested for validity include:

- Soldering Machine 1 is better than Soldering Machine 2
- New method can produce higher Output
- The new chemicals are safe and usable

**P-Value Overview**

In hypothesis testing, the p-value is used to evaluate the probability that the results of a study occurred by chance. The p-value is calculated based on the data and the assumptions of the null hypothesis.

A lower p-value indicates that it is less likely that the results occurred by chance, and as a result, the null hypothesis is more likely to be rejected.

When interpreting a p-value, a common threshold is 0.05. If the p-value is less than 0.05, it is considered statistically significant, and the null hypothesis is rejected in favor of the alternative hypothesis.

Conversely, if the p-value is greater than 0.05, it is not considered statistically significant, and the null hypothesis is not rejected. It is important to note that a p-value of 0.05 is not a hard rule and it can be adjusted based on the specific context of the study.

- If the p-value is less than 0.05, the result is statistically significant. In this case, you reject the null hypothesis favoring the alternative hypothesis.
- If the p-value is greater than 0.05, then the result is not statistically significant and hence doesn’t reject the null hypothesis.

**Z-Test Definition**

A z-test is a statistical test used to determine if the sample mean differs from a population mean. It is used to evaluate the probability that the difference between the sample mean and the population mean is due to chance.

The z-test assumes that the population standard deviation is known and the sample size is large (typically greater than 30). It uses the standard normal distribution (z-distribution) to calculate the probability or p-value associated with the sample mean.

Depending on the p-value, the null hypothesis (that the sample mean is equal to the population mean) is either rejected or not rejected.

### Example

Let’s say that the mean score of students in a class is greater than 70 with a standard deviation of 10. If a sample of 50 students was selected with a mean score of 80, calculate the Z-value to check if there is enough evidence to support this claim at a 0.05 significance level.

#### Solution:

Here, the sample size is 50 and we know the standard deviation. This is a case of a right-tailed one-sample z test.

The Null hypothesis is the mean score is 70

The Alternative hypothesis is mean score is greater than 70

From the z-table, the critical value at alpha = 0.05 is 1.645

Xbar = 80

μ = 70

n = 50

σ = 10

Substituting the values in the formula, you will get the Z value to be equal to 7.09.

Since 7.09 > 1.645 thus, the null hypothesis is rejected and there is enough to support that the mean of the class is greater than 70.

**T-Test Definition**

A t-test is a statistical test used to determine whether there is a significant difference between the means of two groups.

It is commonly used to compare the means of two groups, such as a control group and a treatment group, or to compare a sample mean to a known population mean. There are two types of t-tests: the one sample t-test and the independent sample t-test.

The t-test is based on the t-distribution, which is a probability distribution that is used to estimate population parameters when the sample size is small or when the population standard deviation is unknown.

The t-test is a parametric test and assumes that the data is normally distributed.

### Example:

A store wants to improve its sales. The previous sales data shows that the average sale of 30 salesmen was $40 per sale. After some training, the current data showed an average sale of $60 per transaction. If the standard deviation given is $20, find the t-value. Did training improve the sales?

#### Solution:

Xbar = 60

s = 20

n = 30

μ = 40

Substituting the values in the formula, you will get t-value = 5.47. For the alpha value of 0.05, the critical value is 1.711. Here 5.47 > 1.711, we can reject the null hypothesis and conclude that training did affect sales.

**Z-Test and T-Test Comparison**

A Z-test is a statistical test used to determine if there is a significant difference between the mean of a population and a specific value, or between the means of two populations. The Z-test assumes that the data being tested is normally distributed and that the population standard deviation is known.

A T-test, on the other hand, is a statistical test used to determine if there is a significant difference between the means of two groups. Unlike the Z-test, the T-test does not assume that the population standard deviation is known and instead uses the sample standard deviation as an estimate. Additionally, T-test is used when sample size is small and/or population standard deviation is unknown.

In summary, both Z-test and T-test are used to compare means of two groups, but Z-test assumes that the population standard deviation is known, while T-test does not.