# Statistical Hypothesis Testing Using R: A Practical Guide

Embark on a journey of statistical exploration with our guide on hypothesis testing using the R programming language. In this tutorial, we demystify the process through two concrete examples: testing coin fairness and performing a chi-square test of association. From formulating hypotheses to interpreting results, each step is meticulously explained, and R code snippets are provided for seamless replication. Whether you're a novice seeking foundational knowledge or a seasoned data analyst honing your R skills, this tutorial equips you with the tools to navigate hypothesis testing confidently in real-world scenarios. Welcome to the intersection of theory and practice in statistical inference using R.

## Problem Description:

The hypothesis testing using the R assignment involves applying the six classical steps in hypothesis testing to two different scenarios. The first scenario deals with testing the fairness of a coin through a proportion test, while the second involves a chi-square test of association to explore potential relationships between categorical variables.

### Example 1: Testing Coin Fairness

H0: p = 0.5 (The coin is fair)

H1: p ≠ 0.5 (The coin is not fair)

Note: this is a two-tailed test since the alternative hypothesis carries ≠ a sign. It can either be greater than 0.5 or less than 0.5

Formulate the Hypothesis:

• Null Hypothesis (H0): =0.5p=0.5 (The coin is fair)
• Alternative Hypothesis (H1): ≠0.5p=0.5 (The coin is not fair)

Specify the Level of Significance:

=0.05α=0.05 (5%)

State the Test Statistic:

• Use Z-test since conditions

Z=(sample statistic-null parameter)/(standard error)

State the Decision Rule:

• Reject the null hypothesis if the p-value is less than 0.05.

Computation of Test Statistic and P-value:

• Sample proportion =0.46p=0.46
• Standard Error 0.07SE(P)=0.07
• Test Statistic −0.571Z=−0.571
• P-value =2⋅(1−0.7157)=0.568=2⋅(1−0.7157)=0.568

Comparison/Decision:

• Since the p-value (0.568) is greater than 0.05, we fail to reject the null hypothesis, concluding that the coin is fair.

### 95% Confidence Interval:

• ±1.96)P±1.96⋅SE(P)
• Confidence Interval: [0.323,0.597][0.323,0.597]

Interpretation: We are 95% confident that the true proportion of success in a coin toss experiment lies between 0.323 and 0.597.

### Example 2: Chi-square Test of Association

Formulate the Hypothesis:

• Null Hypothesis (H0): There is no association between the row and column variables.

Specify the Level of Significance:

• 0.05α=0.05 (5%)

State the Test Statistic:

• 2=∑(2χ2=∑E(O−E)2, df=(−1)⋅(−1)df=(r−1)⋅(c−1)

State the Decision Rule:

• Reject the null hypothesis if the p-value is less than 0.05.

Computation of Test Statistic and P-value:

• Compute 2χ2 and find the p-value using a chi-square table or statistical software.

Comparison/Decision:

• Compare the p-value with 0.05 and conclude.

### Share and Discuss Sample Size and Variation:

• Bigger Sample: Increase in test statistic, decrease in p-value.
• Smaller Sample: Decrease in test statistic, increase in p-value.
• Similar Group Values: Small test statistic, high p-value.
• Different Group Values: Large test statistic, low p-value.