Hypothesis testing By Dr. Ama

Hypothesis Testing

• Definition: An act in statistics whereby an analyst tests an assumption regarding a population parameter. (Source: Investopedia)

• Instructor: Dr. Ama Jayawardana, Senior Lecturer, General Sir John Kotelawala Defense University.

Fundamentals of Hypothesis Testing

• Hypothesis testing is a technique for interpreting and drawing inferences about a population based on sample data.

• It helps determine which sample data best support mutually exclusive population claims.

Hypotheses

• Null Hypothesis (H0): The assumption that the event will not occur. It has no impact unless rejected.

  • Symbol: H0 (pronounced H-naught)

• Alternate Hypothesis (H1 or Ha): The logical opposite of the null hypothesis.

t Tests

• Definition: A t test is a statistical test used to compare the means of two groups.

• Purpose: Used in hypothesis testing to determine if a treatment has an effect on the population or if two groups differ significantly.

t Test Example

• Scenario: Testing if the mean petal length of iris flowers differs by species.

• Procedure: Measure 25 petals from two different species and apply a t test with null and alternative hypotheses.

  • H0: The true difference between group means is zero.

  • Ha: The true difference is not zero.

When to Use a t Test

• Conditions:

  • Data must be independent and approximately normally distributed.

  • Groups must exhibit homogeneity of variance (similar variance).

t Test Formula

t = (x1 - x2) / (s2 / n1 + s2 / n2)

• Where:

  • t = test statistic

  • x1, x2 = means of the groups

  • s2 = pooled standard error

  • n1, n2 = number of samples in each group.

Sample Data for t Test

Calculating Differences

• Difference in averages: (22.29 - 14.95) = 7.34.

Pooled Standard Deviation

1. Calculate pooled variance: For example, use:

   • 816.55 / 21 = √38.88 = 6.24.

Test Statistic Calculation

• Combine calculated values:

• Final formula used:

  • t = 7.34 / (√(6.24 × (1/10 + 1/13))) = 2.80.

Finding Degrees of Freedom and Alpha Level

• Degrees of Freedom (df): n1 + n2 - 2 = 21.

• Significance level (α = 0.05). The critical t value for df=21 is 2.080.

Conclusion of t Test

• Compare t value to critical t value: If t > 2.080, reject H0.

• Conclusion: Significant difference in body fat between men and women.

Chi-square Test

• Definition: A statistical procedure to determine the difference between observed and expected data, or to examine the relationship between categorical variables.

Types of Chi-Square Tests

1. Independence: Examines if two categorical variables are related.

2. Goodness-of-Fit: Determines if a variable is likely from a certain distribution.

Steps for Chi-Square Test of Independence

• Null Hypothesis: There is no relationship between variables.

• Calculate expected values based on the sample distribution and total counts.

• Calculate the Chi-square statistic.

Example Data

• Gender and education levels across a survey of 395 individuals.

Calculation of Test Statistic

• Use the formula: χ² = Σ(O−E)²/E.

• Compare χ² value to critical value to determine result significance.

One-Way ANOVA

• Definition: A statistical method used to compare means across multiple groups.

• Key Points:

  • Dependent variable must be continuous.

  • Independent variable must be categorical.

  • Assumes normality and homogeneity of variance.

Process Steps for One-Way ANOVA

1. Calculate group means and overall mean.

2. Calculate SSR (Regression Sum of Squares) and SSE (Error Sum of Squares).

3. Fill in ANOVA table using totals obtained.

4. Compare F statistic to critical F value to draw conclusions.

Example Interpretation

• F test statistic results and comparing to critical F values guide null hypothesis acceptance or rejection.

Practical Application

• Chi-square and t tests are steps to assess relationships and effects in real-world scenarios, like analyzing apple weights or political preferences.

Conclusion

• Research methods in statistics like t tests and Chi-square tests are vital tools in drawing inferences from data.