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Question 7
In the country of United States of Heightlandia, the height measurements of ten-year-old children are approximately normally distributed with a mean of 53.6 inches, and standard deviation of 3.3 inches.
Answer the following, rounding your answers to three decimal places where necessary.
a. What is the probability that a randomly chosen child has a height of less than 58.85 inches? Answer =
b. What is the probability that a randomly chosen child has a height of more than 50.4 inches? Answer $=$
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Final Answer: The probability that a randomly chosen child has a height of less than 58.85 inches is \( \boxed{0.944} \). The probability that a randomly chosen child has a height of more than 50.4 inches is \( \boxed{0.834} \).
Step 1 ：Given that the height measurements of ten-year-old children in the country of United States of Heightlandia are approximately normally distributed with a mean of 53.6 inches, and standard deviation of 3.3 inches.
Step 2 ：We are asked to find the probability that a randomly chosen child has a height of less than 58.85 inches and the probability that a randomly chosen child has a height of more than 50.4 inches.
Step 3 ：We first calculate the z-scores for the given heights. The z-score is a measure of how many standard deviations an element is from the mean. It is calculated as \( z = \frac{X - \mu}{\sigma} \), where \( X \) is the element, \( \mu \) is the mean, and \( \sigma \) is the standard deviation.
Step 4 ：For a height of 58.85 inches, the z-score is \( z_a = \frac{58.85 - 53.6}{3.3} = 1.590909090909091 \).
Step 5 ：For a height of 50.4 inches, the z-score is \( z_b = \frac{50.4 - 53.6}{3.3} = -0.9696969696969706 \).
Step 6 ：We then use the cumulative distribution function (CDF) to find the probabilities. The CDF at a point is the probability that a random variable takes a value less than or equal to that point.
Step 7 ：The probability that a randomly chosen child has a height of less than 58.85 inches is the CDF at \( z_a \), which is 0.944.
Step 8 ：The probability that a randomly chosen child has a height of more than 50.4 inches is 1 minus the CDF at \( z_b \), which is 0.834.
Step 9 ：Final Answer: The probability that a randomly chosen child has a height of less than 58.85 inches is \( \boxed{0.944} \). The probability that a randomly chosen child has a height of more than 50.4 inches is \( \boxed{0.834} \).
