STATS

Chapter 4: Hypothesis Testing

Objective: Hypothesis Statistics

The primary objective of hypothesis testing is to determine if there is a statistically significant difference in mean scores based on sex classification. Understanding these differences is crucial in various fields, including psychology, medicine, and social sciences.

Hypotheses

• Null Hypothesis (H0): Assumes no significant difference exists in mean scores between sexes, suggesting any observed difference is due to random chance.

• Alternative Hypothesis (Ha): Proposes that there is a significant difference in mean scores between sexes, indicating a real effect.

Steps in Hypothesis Testing:

1. Formulate Null and Alternative Hypotheses: Clearly define both hypotheses to guide the testing process.

2. Set Significance Level (α): Choose a threshold (commonly 0.05) indicating the probability of committing a Type I error (rejecting the null hypothesis when it is true).

3. Determine Appropriate Test Statistic: Depending on the data type and sample size, select either a t-test for smaller samples or ANOVA for larger sets involving multiple groups.

4. Requisite Conclusion: Calculate the test statistic from sample data and compare it against the critical value or the p-value against α.

5. State the Conclusion: Based on the comparison, either reject the null hypothesis if evidence suggests a significant difference or fail to reject it if the evidence is insufficient.

Significance Level (α)

The significance level reflects the acceptable error magnitude in rejecting the null hypothesis. A lower α value denotes a stricter criterion for significance. Common levels are 0.01, 0.05, and 0.10.

Types of Hypothesis Tests

• Null Hypothesis (H0): This represents the status quo, claiming no significant difference or relationship exists.

• Alternative Hypothesis (Ha): Opposes H0, indicating a significant difference or relationship may exist.

Tests Used

• ANOVA (F-test): Essential when comparing means across three or more groups, allowing for variance analysis among group means.

• Chi-Square Test: Suitable for categorical variables, assessing differences in distributions between groups.

• Correlation: Utilizes Pearson's r to measure the strength and direction of the relationship between two variables.

Statistical Testing Techniques

Z Test & T Test:

• Z Test: Applied for comparing means in independent groups typically following a normal distribution with a known population variance.

• T Test: Used for dependent or matched groups, helpful for smaller sample sizes where population variance is unknown.

Chapter 5: Testing Means of Two Independent Groups

Assumptions and Usage of Tests

T-tests determine differences between means of two independent groups but rely on specific assumptions such as normality of data and equality of variances.

One Tailed vs. Two Tailed Tests

• One Tailed Test: This directional test assesses whether one group’s mean is significantly greater than the other.

• Two Tailed Test: Non-directional, it tests for any significant difference between groups without specifying a direction.

Example Testing Mean Differences

Subject: Birth Length of Male vs. Female Infants

• H0: There is no significant difference in mean birth lengths between genders.

• Ha: Male infants exhibit greater mean birth lengths than female infants.

Page 2: Comparing Proportions with Z Test

Example 1: Proportion Test

A researcher examines endurance test success between smokers and non-smokers, testing hypotheses about success proportions.

• Hypothesis Testing:

  • H0: The success proportions between smokers and non-smokers are equal.

  • Ha: There is an unequal success proportion between the two groups.

  • The researcher must calculate critical values and test statistics, proceeding to reject or not reject H0.

Two-Tailed Test Example

If the computed Z value exceeds the critical Z value, H0 is rejected, and the conclusion addresses the relationship between smoking habits and performance outcomes.

Page 3: Non-Parametric Statistical Tests

Chi-Square Test

This test assesses whether the distributions of two groups are equal. For example, comparing television viewing preferences among favorable and unfavorable groups provides insights into patterns of behavior.

Mann-Whitney Rank Sum Test

A non-parametric alternative to t-tests, specifically designed for ordinal data. This test evaluates differences between independent groups assuming no normal distribution conditions.

Conclusion on Relationships

The chi-square test reveals the strength of relationships and group differences, effectively summarizing data on categorical variables and aiding in decision-making processes in research.