• Definition: The p-value is a probability measure that helps determine the significance of the results obtained from a statistical test. It quantifies the evidence against the null hypothesis ( H_0 ).
• Range: The p-value ranges between 0 and 1.
• Interpretation:
  • Low p-value (typically < α): Strong evidence against the null hypothesis, leading to rejection of ( H_0 ) in favor of ( H_1 ).
  • High p-value (typically ≥ α): Weak evidence against the null hypothesis, leading to failure to reject ( H_0 ).
• Key Points:
  • Significance Level: Choose a significance level (𝛼) before conducting the test (commonly 0.05).
  • Decision Rule: If p-value < α, reject ( H_0 ), if p-value ≥ α, fail to reject ( H_0 ).

1. Formulate Hypotheses:
   • Null Hypothesis ( H_0 ): No effect or no difference.
   • Alternative Hypothesis ( H_1 ): Some effect or difference.
2. Choose Significance Level ( alpha ): Commonly set at 0.05, 0.01, or 0.10.
3. Conduct the Statistical Test: Calculate the test statistic and the corresponding p-value.
4. Compare p-Value with α:
   • If p-value < alpha : Reject ( H_0 ). The results are statistically significant.
   • If p-value ≥ alpha : Fail to reject ( H_0 ). The results are not statistically significant.

• Definition: The null hypothesis states that there is no effect or no difference. It serves as the default or starting assumption.
• Example: ( H_0 ) : The mean score of two groups is equal.

• Definition: The alternative hypothesis states that there is an effect or a difference. It is what the researcher aims to prove.
• Example: ( H_1 ) : The mean score of two groups is different.

1. Choose the Appropriate Test: Depending on the data type and research question (e.g., t-test for means, chi-square test for independence).
2. Set Up the Hypotheses: Null and alternative hypotheses based on the research question.
3. Calculate the Test Statistic and p-Value: Use statistical software or libraries like SciPy in Python.
4. Make a Decision:
   • Compare the p-value with the chosen significance level alpha .

## Common Tests:
## t-tests: Compare means of two groups.
## ANOVA: Compare means of more than two groups.
## Chi-square test: Assess independence between categorical variables.
## Correlation tests: Assess relationships between variables 
## (e.g., Pearson correlation, Spearman correlation, Partial correlation).
from scipy.stats import ttest_ind

## Sample data
group1 = [23, 21, 18, 30, 28]
group2 = [25, 27, 22, 31, 33]

## Conduct t-test
t_stat, p_value = ttest_ind(group1, group2)

## Print results
print(f"t-statistic: {t_stat}, p-value: {p_value}")

## Interpret p-value
alpha = 0.05
if p_value <= alpha:
    print("Reject the null hypothesis. There is a significant difference between the groups.")
else:
    print("Fail to reject the null hypothesis. There is no significant difference between the groups.")  

• The p-value helps determine the statistical significance of test results.
• Null Hypothesis ( H_0 ): Assumes no effect or no difference.
• Alternative Hypothesis ( H_1 ): Assumes some effect or difference.
• Significance Level ( alpha ): Threshold for deciding whether to reject H_0 .
• Use statistical tests to calculate p-values and make informed decisions based on the data.

By understanding these concepts and following the outlined steps, you can effectively use p-values and statistical tests to evaluate hypotheses and make data-driven decisions.

There are various distribution but the major distribution used in data science are :

Statistical Distributions
Family of Distribution