Notes on Non-Parametric Testing of Hypothesis: Sign Test

Lecture Overview

• Instructor: Dr. Gaur, School of Mathematics, Thapar Institute, India
• Topic: Non-parametric testing of hypothesis - Sign test
• Contact: Email or YouTube channel for queries

Introduction to Non-Parametric Tests

• Definition: Non-parametric tests do not assume a normal distribution in the population.
• Importance: Used when data does not meet the assumptions required for parametric tests (e.g., Z test, T test, ANOVA).
• Examples of Non-Parametric Tests:
  • Sign test
  • Wilkerson test
  • Mann-Whitney test
  • Kruskal-Wallis test
  • Rank correlation

What is the Sign Test?

• Nature of the Test: One of the simplest non-parametric tests that focuses on the signs (+ or -) of deviations rather than their magnitudes.
• Application: Used to test hypotheses about the median of a single population (single sample).
• Hypotheses:
  • Null hypothesis (H0): Median (η) equals a specified value (η0).
  • Alternative hypothesis (H1): Median is different from the specified value (two-tailed or one-tailed).

Procedure for the Sign Test

1. Define Hypotheses:
   • H0: η = η0
   • H1: η ≠ η0 (or ≤, ≥ for one-tailed tests)
2. Data Collection:
   • Assume a sample size (n) from the population.
3. Calculation of Deviation Signs:
   • Subtract η0 from each observation.
   • Record signs:
     • Positive (+) for positive deviations
     • Negative (-) for negative deviations
     • Zero (0) for zero deviations (discarded later)
4. Count Signs:
   • Let T+ = number of positive signs
   • Let T- = number of negative signs
   • T = minimum(T+, T-)

Example Calculation for Small Samples (n < 25)

• Step 1: Define the null and alternative hypotheses.
• Step 2: Calculate T+ and T-.
• Step 3: Critical value (Tc) is obtained from the table based on sample size and significance level.
• Step 4: Critical region is defined where T ≤ Tc leads to rejection of H0.

Examples of Sign Test in Practice

1. Example A: Study hours before a statistics test.

   • Hypotheses defined.
   • Deviation signs calculated, T+ = 8, T- = 3.
   • Critical value look up for n=11, Tc = 1.
   • Conclusion: Fail to reject H0, median = 3.

2. Example B: Teacher’s claim about study time.

   • Calculate T+ and T-.
   • Use critical value to assess H0.

Large Samples (n > 25)

• Use normal approximation to the binomial distribution.
• Calculate Z-test statistic for decision making.
• Compare Z-value to critical values for hypothesis testing.

Summary of Steps for Sign Test

1. Define H0 and H1.
2. Calculate deviation signs and counts T+ and T-.
3. Compare with critical values or calculate Z value and use p-values.
4. Make a decision: reject or accept H0 based on the calculated values.

Conclusion

• The sign test is a straightforward method for hypothesis testing when normality cannot be assumed.
• Next Lecture: Wilkerson Test.
• Note: For further reading, students are encouraged to explore various statistics resources and subscribe to the channel.