# Master Hypothesis Testing with One Proportion Z-Test in Python

## One Proportion Z-Test

Do you ever wonder if your hypothesis about a certain proportion is valid? One way to answer that question is through a One Proportion Z-Test.

## Null and Alternative Hypotheses

### 1. Null Hypothesis

The Null Hypothesis is the statement that assumes there is no significant difference or relationship between the variables being tested.

### 2. Alternative Hypothesis

On the other hand, the Alternative Hypothesis states that a significant difference exists. In a One Proportion Z-Test, the Null Hypothesis assumes that the proportion is equal to a given value, while the Alternative Hypothesis assumes that the proportion is different from the given value.

## Test Statistic Formula

### 1. Formula

The next step is to compute for the Test Statistic using the formula:

``z = (p - P) / sqrt(P*(1-P)/n)``

### 2. Variables

• p is the observed sample proportion
• P is the hypothesized population proportion
• n is the sample size

The value of z will be computed based on the difference between the observed and hypothesized proportions, as well as the sample size.

## Interpretation of Results

### 1. p-value

After computing for the Test Statistic, the next step is to interpret the results. The p-value is obtained from the Z-Table, which can be compared to a predetermined Significance Level.

### 2. Decision

• If the p-value is less than the Significance Level, then we reject the Null Hypothesis in favor of the Alternative Hypothesis.
• On the other hand, if the p-value is greater than the Significance Level, then we fail to reject the Null Hypothesis.

## Python Implementation

### 1. Function Syntax

Now that we understand the theory behind the One Proportion Z-Test, let us see how it can be implemented in Python using the `proportions_ztest()` function.

The basic syntax of the `proportions_ztest()` function is:

``proportions_ztest(count, nobs, value=None, alternative='two-sided', prop_var=False)``

### 2. Parameters

• `count` is the number of successes in the sample
• `nobs` is the sample size
• `value` is the hypothesized proportion (default value is 0.5)
• `alternative` is the type of Alternative Hypothesis (default is ‘two-sided’)
• `prop_var` is a boolean value that determines the use of the sample proportion or hypothesized proportion (default is False)

### 3. Example Scenario

Suppose we want to test the hypothesis that the proportion of residents in a certain city who support a particular law is equal to 0.6. We surveyed 500 residents and found that 320 of them support the law. To test this hypothesis using the `proportions_ztest()` function, we can input the following values:

• `count` = 320
• `nobs` = 500
• `value` = 0.6

### 4. Output Interpretation

The output of the `proportions_ztest()` function will provide us with the Z-Score and the p-value. If we set the Significance Level to 0.05, then we can interpret the output as follows:

• Z-Score: 1.6974
• p-value: 0.0894

Since the p-value is greater than the Significance Level, we fail to reject the Null Hypothesis.

We can therefore conclude that there is not enough evidence to support the claim that the proportion of residents who support the law is different from 0.6.

In conclusion, the One Proportion Z-Test is a useful statistical tool in hypothesis testing. By formulating Null and Alternative Hypotheses, computing for the Test Statistic, and interpreting the results based on a predetermined Significance Level, we can determine whether or not our hypothesis is valid.

With the help of the `proportions_ztest()` function in Python, implementing the One Proportion Z-Test is made easier and more efficient. In summary, the One Proportion Z-Test is a statistical method used to test the null and alternative hypotheses for proportion-related tests.

The test statistic formula and interpretation of the results are crucial in understanding the One Proportion Z-Test. Additionally, the `proportions_ztest()` function in Python simplifies the process by providing a quick output of the Z-Score and p-value.

It is essential to correctly formulate the Null and Alternative Hypotheses and determine the Significance Level before interpreting the results. By knowing the basics of the One Proportion Z-Test and being able to implement it in Python, one gains a better understanding of how to perform hypothesis testing accurately.