A GLIMPSE INTO P-VALUES

A p-value measures how compatible your data is with the null hypothesis.

• Small p-values suggest that data is unlikely if H₀ is true.
• Large p-values suggest that data is consistent with H₀.

Hypothesis testing compares two statements:

• Null Hypothesis (H₀): No effect or no difference.
• Alternative Hypothesis (H₁): There is an effect or difference.

The p-value helps in deciding if H₀ should be rejected.

The p-value is the probability of observing data at least as extreme as the sample data if the null is true:

\[ p\text{-value} = P(\text{data as extreme as observed} \mid H_0 \text{ true}) \]

• p-value < α (e.g., 0.05) → reject H₀.
• p-value ≥ α → fail to reject H₀.
• A small p-value does not prove H₁ it just suggests that H₀ is unlikely.

• H₀: Die is fair (P(6) = 1/6)
  
• H₁: Here, Die is biased
  
• Roll the die 60 times and count sixes.

## [1] 9

• Assume that die_rolls is defined from previous example
• We use Binomial test to check if number of sixes is consistent with a fair die

## 
##  Exact binomial test
## 
## data:  sum(die_rolls) and length(die_rolls)
## number of successes = 9, number of trials = 60, p-value = 0.863
## alternative hypothesis: true probability of success is not equal to 0.1666667
## 95 percent confidence interval:
##  0.0709562 0.2657404
## sample estimates:
## probability of success 
##                   0.15

• P-values measure how fascinating the data is under the null hypothesis.
• They also guide hypothesis testing but don’t really confirm the alternative hypothesis.
• Visualizations like histograms, boxplots, and interactive 3D plots help understand the p-value behavior.