1. **State the problem:** We want to find the probability that a randomly chosen ten-year-old child in Heightlandia has a height between 53.05 and 54.95 inches. 2. **Given data:** The heights are normally distributed with mean $\mu = 56.4$ inches and standard deviation $\sigma = 1.7$ inches. 3. **Formula:** For a normal distribution, the probability that $X$ lies between $a$ and $b$ is $$P(a < X < b) = P\left(\frac{a - \mu}{\sigma} < Z < \frac{b - \mu}{\sigma}\right)$$ where $Z$ is a standard normal variable with mean 0 and standard deviation 1. 4. **Calculate the z-scores:** $$z_1 = \frac{53.05 - 56.4}{1.7} = \frac{\cancel{53.05} - 56.4}{\cancel{1.7}} = \frac{-3.35}{1.7} = -1.970588235$$ $$z_2 = \frac{54.95 - 56.4}{1.7} = \frac{\cancel{54.95} - 56.4}{\cancel{1.7}} = \frac{-1.45}{1.7} = -0.852941176$$ 5. **Find the probabilities from the standard normal table or using a calculator:** $$P(Z < z_1) = P(Z < -1.9706) \approx 0.0244$$ $$P(Z < z_2) = P(Z < -0.8529) \approx 0.1967$$ 6. **Calculate the probability between the two z-scores:** $$P(z_1 < Z < z_2) = P(Z < z_2) - P(Z < z_1) = 0.1967 - 0.0244 = 0.1723$$ 7. **Final answer:** The probability that a randomly chosen child has a height between 53.05 and 54.95 inches is approximately **0.172** (rounded to 3 decimal places).