To find the probability that the height of a randomly chosen ten-year-old child in Heightlandia is between 46.9 and 57.4 inches, we can use the properties of the normal distribution. Given:
- Mean (\(\mu\)) = 54.9 inches
- Standard deviation (\(\sigma\)) = 4.8 inches
We will use the z-score formula to standardize the values:
\[ z = \frac{(X - \mu)}{\sigma} \]
Step 1: Calculate the z-scores for both bounds.
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For \(X = 46.9\): \[ z_1 = \frac{(46.9 - 54.9)}{4.8} = \frac{-8}{4.8} \approx -1.66667 \]
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For \(X = 57.4\): \[ z_2 = \frac{(57.4 - 54.9)}{4.8} = \frac{2.5}{4.8} \approx 0.52083 \]
Step 2: Use the z-table or a calculator to find the corresponding probabilities.
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Look up \(z_1 \approx -1.66667\): Using z-tables or a normal distribution calculator, we find: \[ P(Z < -1.66667) \approx 0.0478 \]
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Look up \(z_2 \approx 0.52083\): \[ P(Z < 0.52083) \approx 0.6944 \]
Step 3: Find the probability that height is between 46.9 and 57.4 inches.
\[ P(46.9 < X < 57.4) = P(Z < 0.52083) - P(Z < -1.66667) \] \[ = 0.6944 - 0.0478 \approx 0.6466 \]
Final Step: Round the result to three decimal places.
The probability that the height of a randomly chosen child is between 46.9 and 57.4 inches is approximately:
\[ \text{Answer} \approx 0.647 \]
Thus, the final answer is: 0.647