1. The problem involves comparing two sample means and their sum of squares to analyze variance or differences. 2. Given data: Sample 1: $N_1=25$, $\bar{x}_1=52$, Sum of squares = 78 Sample 2: $N_2=49$, $\bar{x}_2=55$, Sum of squares = 34 3. To analyze, we might calculate the pooled variance or test the difference between means. 4. The formula for pooled variance $S_p^2$ is: $$S_p^2 = \frac{(N_1 - 1)S_1^2 + (N_2 - 1)S_2^2}{N_1 + N_2 - 2}$$ where $S_i^2 = \frac{\text{Sum of squares}_i}{N_i - 1}$ 5. Calculate $S_1^2$: $$S_1^2 = \frac{78}{25 - 1} = \frac{78}{24} = 3.25$$ 6. Calculate $S_2^2$: $$S_2^2 = \frac{34}{49 - 1} = \frac{34}{48} \approx 0.7083$$ 7. Calculate pooled variance: $$S_p^2 = \frac{(25 - 1) \times 3.25 + (49 - 1) \times 0.7083}{25 + 49 - 2} = \frac{24 \times 3.25 + 48 \times 0.7083}{72}$$ $$= \frac{78 + 34}{72} = \frac{112}{72} \approx 1.5556$$ 8. The pooled standard deviation is: $$S_p = \sqrt{1.5556} \approx 1.2472$$ 9. To test difference between means, use the t-statistic: $$t = \frac{\bar{x}_1 - \bar{x}_2}{S_p \sqrt{\frac{1}{N_1} + \frac{1}{N_2}}} = \frac{52 - 55}{1.2472 \sqrt{\frac{1}{25} + \frac{1}{49}}}$$ 10. Calculate denominator: $$\sqrt{\frac{1}{25} + \frac{1}{49}} = \sqrt{0.04 + 0.02041} = \sqrt{0.06041} \approx 0.2458$$ 11. Calculate t: $$t = \frac{-3}{1.2472 \times 0.2458} = \frac{-3}{0.3067} \approx -9.78$$ 12. This large magnitude of t suggests a significant difference between the sample means. Final answer: The t-statistic comparing the two sample means is approximately $-9.78$, indicating a significant difference between the samples.