Study Notes on T-Tests and Statistical Analysis

Introduction to T-Tests

• T-tests are statistical tests used to determine if there are significant differences between the means of two groups.

• The t-score is calculated using a similar formula to the z-score but uses estimated standard error instead of the actual standard error of the mean (SEM).

Calculation of the T-Score

• The formula for calculating the t-score is: $t = \frac{\bar{x} - \mu_0}{SE}$ where:

  • $\bar{x}$ = sample mean

  • $\mu_0$ = population mean (in null hypothesis)

  • $SE$ = estimated standard error of the mean

• Example:

  • For a sample mean of 13 seconds, a population mean of 10 seconds, and an estimated standard error (SE), the t-score was calculated as:

  • $t = \frac{13 - 10}{SE}$ resulting in a t-score of 2.13.

Null Hypothesis Testing

• The null hypothesis (H0) can be stated as follows: There is no significant difference in the time newborns look at their parents' photo compared to a stranger's photo.

• The observed t-value was 2.13, whereas the critical t-value from the t-distribution table (based on degrees of freedom and alpha level) was 2.306.

  • Since the observed t-value (2.13) is less than the critical t-value, we fail to reject the null hypothesis.

  • This implies that, based on this data, we cannot conclude that newborns spent significantly more time looking at their parents' photo than at a stranger's photo.

Interpretation of Results

• The terminology can be confusing, but failing to reject the null hypothesis does not prove it is true; rather, it indicates insufficient evidence to support the alternative hypothesis.

• We fall short of demonstrating a significant effect because our alpha level (0.05) was not met.

• It is crucial to report results even when they are not significant.

  • Format for reporting non-significant results:

  • Indicate the p-value (p > 0.05) since it signifies the proportion of the tails.

  • Report means and standard errors (SE) in the results.

Reporting T-Test Results

• When reporting t-values in text, include the degrees of freedom (df), as they affect the critical value.

• Example format:

  • "A one-sample t-test revealed that newborns spent an average of 13 seconds looking at a photo of their parents compared to 10 seconds. The t-value was 2.13 (df = 8), which is not significant at the alpha level of 0.05."

Increasing Sample Size and Its Effects

• In subsequent experiments, the sample size was increased from 9 to 20 newborns.

• With 20 newborns, the average time spent looking at their parents remained at 13 seconds versus the stranger’s photo.

• The hypothesis remained the same (H0: no difference in time spent).

  • Critical t-value now: 2.093 (degrees of freedom = 19, calculated as 20 - 1).

• The new t-value was calculated to be 3.16, allowing for the rejection of the null hypothesis since 3.16 > 2.093.

  • Conclusion: Newborns spent significantly longer looking at their parents' faces compared to strangers.

Understanding Standard Error Changes

• An increase in sample size decreases the standard error due to the relationship in variance

  • Standard Error is influenced by the size of the sample:
    $SE = \frac{S}{\sqrt{n}}$

• Consequently, with the previous standard error of 1.41 decreasing to 0.95 for a new sample size, confidence in the t-test increases.

Effect Size Measures

• Cohen's d is computed using the sample mean versus the null mean, where: $d = \frac{\bar{x} - \mu_0}{S}$

  • Here, S represents the standard deviation of the sample.

• R-squared represents the proportion of variance explained by the independent variable:

  • Formula:
    $R^2 = \frac{t^2}{t^2 + df}$

  • R-squared values range from 0 to 1 and indicate the percentage of variance explained by the independent variable.

Interpretation of R-squared

• A hypothetical R-squared value of 0.34 means that 34% of the variance in how long babies look at their parents' photo compared to a stranger's is explained by that independent variable.

• The remaining 66% of the variance could be explained by other factors and considerations.

• This emphasizes the complexity of human behavior and the multitude of influences affecting outcomes in psychological studies.

Conclusion

• It’s crucial to balance sample size and significance levels (alpha) to robustly detect effects.

• Larger sample sizes can reduce error and provide a clearer picture of the differences observed between groups.

• Researchers must remain cautious in interpreting findings, as variance from multiple sources can complicate determinations of causality.