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Type II Error in R, we will discuss the concept of Errors in Hypothesis Testing, explore the different types of errors that may arise during this process, and learn how to calculate them.

Hypothesis testing involves forming assumptions about a model or data set, which serve as the foundation for further analysis.

The term ‘Error’ in Hypothesis Testing refers to the estimation of accepting or rejecting a specific hypothesis.

Two primary types of errors can occur in this context:

1. Type I Error (Alpha Error): This error happens when we reject the Null Hypothesis, even though it is indeed true. It is also referred to as a ‘false positive,’ as it leads to incorrectly rejecting a correct hypothesis.
2. Type II Error (Beta Error): This error occurs when we fail to reject the Null Hypothesis despite it being incorrect, and the Alternative Hypothesis is correct. It is known as a ‘false negative,’ as it results in retaining an incorrect hypothesis.

Understanding and managing these errors is crucial for conducting accurate hypothesis testing and drawing valid conclusions from the data.

Note:
P(X) is the probability of the event X happening.
Ho = NULL Hypothesis
Ha = Alternative Hypothesis

In this example, we will examine a jury/court decision-making process for a case, where the jury has two primary options: determining the convict is guilty or not guilty. These two decisions correspond to the two hypotheses considered in hypothesis testing. For every decision made by the jury, the reality can be either:

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1. The convict is genuinely guilty in real life, and the jury’s decision is that they are guilty.
2. The convict is not guilty in real life, and the jury’s decision is that they are not guilty.

Now, let’s explore the two types of errors that can arise in this context:

1. Type I Error (False Conviction): This error occurs when the jury decides the convict is guilty, but in reality, they are not guilty. It results in an incorrect conviction and can have severe consequences for the individual’s life and future.
2. Type II Error (Missed Conviction): This error happens when the jury decides the convict is not guilty, even though they are genuinely guilty in reality. It leads to an incorrect acquittal, potentially allowing a criminal to go free and continue committing crimes.

Minimizing these errors is crucial for maintaining justice and ensuring fair verdicts in court proceedings.

Ho = Not Guilty
Ha = Guilty

#In this example, 

Type I Error will be: Innocent in Jail
Type II Error will be: Guilty Set Free

To Calculate the Type II Error in R Programming:

Type II Error, also known as Beta Error, can be determined using a specific formula. However, in this article, we will demonstrate how to calculate Type II Error using the R programming language for enhanced efficiency and accuracy.

P(Probability of failing to remove Ho / Probability of Ho being false ) = P(Accept Ho | Ho False)
library(pwr)
# Parameters
effect_size <-0.5 # The difference between null and alternative hypotheses
sample_size <- 30 # The number of observations in each group
sd <- 4 # The standard deviation
alpha <- 0.05 # The significance level

# Calculate Type II Error
pwr_result <-pwr.t.test(n = sample_size,d = effect_size / sd,sig.level = alpha,type = "two.sample",alternative = "two.sided")
type_II_error <- 1 - pwr_result$power
# Print Type II Error
print(type_II_error)
0.9236474

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