1. **Problem statement:** Transform the given sequence of random numbers uniformly distributed on [0,1] to a uniform distribution on [5,10], then find the mean of the transformed sample and the true mean of the uniform distribution on [5,10]. 2. **Transforming the sequence:** Given original numbers $x_i$ uniformly distributed on [0,1], the transformation to $y_i$ uniformly distributed on [5,10] is: $$y_i = 5 + 5x_i$$ (since the length of the new interval is $10 - 5 = 5$). 3. **Apply the transformation:** For each $x_i$: - $0.038 \to 5 + 5 \times 0.038 = 5 + 0.19 = 5.19$ - $0.931 \to 5 + 5 \times 0.931 = 5 + 4.655 = 9.655$ - $0.538 \to 5 + 5 \times 0.538 = 5 + 2.69 = 7.69$ - $0.153 \to 5 + 5 \times 0.153 = 5 + 0.765 = 5.765$ - $0.391 \to 5 + 5 \times 0.391 = 5 + 1.955 = 6.955$ - $0.780 \to 5 + 5 \times 0.780 = 5 + 3.9 = 8.9$ 4. **Calculate the mean of the transformed sample:** $$\bar{y} = \frac{5.19 + 9.655 + 7.69 + 5.765 + 6.955 + 8.9}{6} = \frac{43.155}{6} = 7.1925$$ 5. **Calculate the true mean of the uniform distribution on [5,10]:** The mean of a uniform distribution on $[a,b]$ is: $$\mu = \frac{a + b}{2} = \frac{5 + 10}{2} = 7.5$$ **Final answers:** - Transformed sequence: $[5.19, 9.655, 7.69, 5.765, 6.955, 8.9]$ - Sample mean: $7.1925$ - True mean: $7.5$