• A p-value helps us evaluate evidence against a null hypothesis.
• It’s the probability of obtaining a result at least as extreme as the one observed, assuming null hypothesis (or \(H_0\)) is true.

\[ H_0: \mu_{\text{High}} = \mu_{\text{Low}} \\ H_A: \mu_{\text{High}} \ne \mu_{\text{Low}} \]

• Set a significance level: \(\alpha = 0.05\)

• Interpret the p-value: \[ \text{If } p < \alpha, \text{ reject } H_0 \]

• Smaller p-values mean stronger evidence against the null hypothesis.

Data used is from the UCI Wine Quality Dataset (Red Wine), which includes:

• Alcohol, pH, acidity, and more
• Quality scored from 0 to 10

Hypothesis: Alcohol content differ significantly between low and high quality wines

Null hypothesis: Alcohol content does not differ between low and high quality wines

• Higher quality alcohol tend to have higher alcohol percentage.

• A majority of low quality alcohol have less than 10% alcohol.
• Alcohol percentage in high quality alcohol is distributed evenly between 9% to 12%.

t.test(alcohol ~ quality_group, data = wine)

• Comparison: Alcohol content between High and Low wine quality groups
  
• Mean Alcohol Content:
  • High Quality: 10.86%
  • Low Quality: 9.93%
• t-value: 19.78
  
• p-value: < 2.2 × 10⁻¹⁶
  
• Conclusion: Significant difference in alcohol content between the two quality groups (p < 0.05)
• 95% CI for difference in means: (0.84, 1.02)

• p-value is 2.2e-16 or 0.00000000000000022
• Since p < 0.05, we reject the null hypothesis.
• There is strong evidence that alcohol content differs between low and high quality wines.
• Higher percentage alcohol is associated with better wine quality.