1. **Stating the problem:** We have a data set with mean $\mu = 7$, variance $\sigma^2 = 2$, and range $R = 3$. Each data point is transformed by multiplying by 2 and then adding 5. We need to determine the truth of the following statements about the new data set: - New mean is 19. - New variance is 10. - New range is 6. 2. **Formulas and rules:** - When each data point $x$ is transformed to $y = a x + b$, the new mean is $\mu_y = a \mu_x + b$. - The new variance is $\sigma_y^2 = a^2 \sigma_x^2$ (adding $b$ does not affect variance). - The new range is $R_y = a R_x$ if $a > 0$ (range scales by the absolute value of $a$). 3. **Calculate the new mean:** $$\mu_y = 2 \times 7 + 5 = 14 + 5 = 19$$ This matches statement 1, so statement 1 is **true**. 4. **Calculate the new variance:** $$\sigma_y^2 = 2^2 \times 2 = 4 \times 2 = 8$$ Statement 2 claims the new variance is 10, but we found 8, so statement 2 is **false**. 5. **Calculate the new range:** $$R_y = 2 \times 3 = 6$$ Statement 3 claims the new range is 6, which is correct, so statement 3 is **true**. **Summary:** - Statement 1: True - Statement 2: False - Statement 3: True However, the user marked statement 2 as true and statement 3 as false, which is incorrect based on calculations.