Statistical Significance and T-Tests

Determining Sample Size and Degrees of Freedom

• To determine where to read on the statistical table, you need to know the sample size.
• The example refers to a previous discussion about a statistically significant difference in birth weight between two groups (milk vs. no milk).
• This scenario represents a two-tailed test with an alpha of 0.05.
• Therefore, you would read from the column corresponding to a two-tailed test with $α = 0.05$.

Degrees of Freedom (DF)

• Degrees of freedom are related to sample size.
• DF represents how many numbers are free to vary.
• For each group in a study, $DF = n - 1$, where n is the sample size of that group.
• The $n - 1$ correction compensates for using sample data.

Example

• If you have five numbers that total 100, four of those numbers can vary freely, but the fifth number is fixed to ensure the total is 100.
• For example, if the first four numbers are 20, 30, 10, and 10 (totaling 70), the fifth number must be 30 to reach 100.

Application to the Study

• In the milk study with two groups (milk and no milk), the degree of freedom is $n - 2$ because there are two groups.
• The experimental group (milk) had $n = 11$ participants, and the control group had $n = 15$ participants.
• The total sample size is $11 + 15 = 26$.
• Therefore, the degree of freedom is $26 - 2 = 24$.
• With $DF = 24$, a two-tailed test, and $α = 0.05$, you can find the critical value on the t-distribution table.

Example 1: Dog Lifespan

Hypothesis

• Research hypothesis: Small dogs live longer than large dogs (one-tailed test).
• Small dog mean lifespan: 28 years.
• Large dog mean lifespan: 26 years.

T-test Results

• Calculated t-value: 2.90 (given).
• Sample size (N): 150.
• Degrees of freedom: $150 - 2 = 148$.
• Alpha level: 0.05.

Determining Statistical Significance

• The hypothesis is one-tailed because it specifies that small dogs live longer.
• The T-critical value from the table (with $DF = 148$ and $α = 0.05$ for a one-tailed test) is 1.65.

Conclusion

• Since the observed t-value (2.90) is greater than the t-critical value (1.65), there is a statistically significant difference in lifespan.
• The observed t-value falls within the zone of rejection.

Role of Statistical Software (e.g., SPSS)

• Researchers typically use software to perform these calculations.
• The researcher specifies the hypothesis (one-tailed or two-tailed).
• The software automatically calculates degrees of freedom and the t-critical value.
• The software compares the observed t-value to the critical value to determine significance.
• The goal is to understand what it means when a study reports statistically significant results.

APA Write-Up for T-Test Results

• This is how statistically significant or non-significant results are typically reported in APA style.
• Example: t(148) = 2.90, p < 0.05
  • t indicates a t-test.
  • 148 is the degrees of freedom.
  • 2.90 is the observed t-value.
  • p is the p-value.

Understanding the P-Value

• The p-value is related to the alpha level.
• If the p-value is less than the alpha level (e.g., 0.05), the result is statistically significant.
• This indicates that the probability of observing the data (or more extreme data) if there were truly no effect is less than 5%.

Complete APA Style Report

• Results indicate that small dogs live statistically longer than large dogs, t(148) = 2.90, p < 0.05.

Additional Elements (not covered here)

• Effect size.
• Statistical power.

Example 2: Weight Change Pill

Hypothesis

• A pill will change your weight.
• This is a two-tailed hypothesis because it does not specify whether the pill will increase or decrease weight.

APA Write-Up for Non-Significant Results

• Example: There is a non-significant mean difference between weight groups, t(14) = 2.20, p > 0.05.

Interpreting the Results

• A degree of freedom of 14 indicates there were 16 individuals in the study (16 - 2 = 14).
• The t-value is 2.20.
• P is greater than 0.05, indicating non-significance.