1. **Problem Statement:** We are given two samples representing annual costs (in thousands of cedis) for public and private universities. We need to summarize the data, formulate and perform a hypothesis test comparing the two means, find critical and p-values, interpret significance, and make a recommendation. 2. **Descriptive Statistics:** Calculate mean and standard deviation for each group. Public University data: $2030, 2200, 2820, 1560, 2410, 2580, 2280, 2580, 1850, 1560, 1400, 2180$ Private University data: $5280, 4320, 4500, 3330, 4400, 3060, 4580, 3780, 5050, 4200, 3600, 4260$ Calculate means: $$\bar{x}_{pub} = \frac{2030 + 2200 + \cdots + 2180}{12} = \frac{26000}{12} \approx 2166.67$$ $$\bar{x}_{pri} = \frac{5280 + 4320 + \cdots + 4260}{12} = \frac{48110}{12} \approx 4009.17$$ Calculate standard deviations $s_{pub}$ and $s_{pri}$ using the formula: $$s = \sqrt{\frac{1}{n-1} \sum (x_i - \bar{x})^2}$$ 3. **Hypothesis Test Formulation:** - Null hypothesis $H_0$: $\mu_{pub} = \mu_{pri}$ (no difference in mean costs) - Alternative hypothesis $H_a$: $\mu_{pub} \neq \mu_{pri}$ (means differ) We use a two-sample t-test assuming unequal variances. 4. **Test Statistic:** $$t = \frac{\bar{x}_{pub} - \bar{x}_{pri}}{\sqrt{\frac{s_{pub}^2}{n_{pub}} + \frac{s_{pri}^2}{n_{pri}}}}$$ Degrees of freedom approximated by Welch-Satterthwaite equation. 5. **Calculate values:** Calculate $s_{pub} \approx 425.88$, $s_{pri} \approx 700.12$ (rounded) Calculate $t$: $$t = \frac{2166.67 - 4009.17}{\sqrt{\frac{425.88^2}{12} + \frac{700.12^2}{12}}} = \frac{-1842.5}{\sqrt{15108.5 + 40857.3}} = \frac{-1842.5}{\sqrt{55965.8}} = \frac{-1842.5}{236.6} \approx -7.79$$ Degrees of freedom $df \approx 18$ (calculated via formula). 6. **Critical value and p-value:** For $\alpha=0.05$ two-tailed test and $df=18$, critical t-value $\approx \pm 2.101$. Since $|t|=7.79 > 2.101$, reject $H_0$. Using t-distribution, p-value $< 0.0001$ (very small). 7. **Statistical Significance:** The test is statistically significant; there is strong evidence that mean annual costs differ between public and private universities. 8. **Recommendation:** GTEC should consider that private universities cost significantly more than public ones. Budget planning should reflect this difference for academic activities.