Applying the Shapiro-Wilk Test

Solution & Explanation

What is the Shapiro-Wilk Test?

The Shapiro-Wilk test is a statistical procedure used to determine if a dataset is normally distributed. It is one of the most powerful tests for normality and is particularly effective for small to medium-sized samples (less than 5000 observations). The test calculates a statistic, W, which quantifies how well the data conforms to a normal distribution.

Purpose of the Shapiro-Wilk Test

The primary purpose of the Shapiro-Wilk test is to assess whether a given sample comes from a normally distributed population. This assessment is crucial when planning to use parametric statistical methods, such as t-tests or ANOVA, which assume normality.

Null Hypothesis

For the Shapiro-Wilk test, the null hypothesis is:

"The sample data is drawn from a population that follows a normal distribution."

The test evaluates whether there is enough evidence to reject this hypothesis.

Conducting the Shapiro-Wilk Test

1. Collect Sample Data: Obtain a sample that you wish to test for normality.
2. Formulate the Null Hypothesis: Assume the data is normally distributed.
3. Perform the Test: Use statistical software or libraries (e.g., Python's scipy.stats.shapiro) to calculate the W statistic and p-value.
4. Interpret the Results:
   • P-value > Significance Level (e.g., 0.05): Fail to reject the null hypothesis, suggesting the data is normally distributed.
   • P-value ≤ Significance Level: Reject the null hypothesis, indicating the data significantly deviates from a normal distribution.

Scenario Example

Imagine a researcher is studying the effects of a new teaching method on student performance. They collect exam scores from two groups of students:

• Group A: Taught using traditional methods.
• Group B: Taught using the new method.

Before applying a t-test to compare the means of the two groups, the researcher needs to verify that the exam scores in each group follow a normal distribution. This is where the Shapiro-Wilk test is applicable.

1. Group A Scores: Conduct the Shapiro-Wilk test.

   • If the p-value is greater than 0.05, assume normality and proceed with the t-test.
   • If not, consider data transformations or non-parametric tests.

2. Group B Scores: Repeat the process.

Conclusion

The Shapiro-Wilk test is a robust tool for assessing normality in datasets, especially when preparing to use parametric statistical methods. By verifying the normal distribution assumption, researchers can ensure the validity and reliability of their statistical analyses.