Question

In the country of United States of Heightlandia, the height measurements of ten-year-old children are approximately normally distributed with a mean of 53.1 inches, and standard deviation of 5.8 inches.

1. What is the probability that a randomly chosen child has a height of less than 52.5 inches? ROUND ANSWER TO 3 DECIMAL PLACES.
   
   
2. What is the probability that a randomly chosen child has a height of more than 53.7 inches? ROUND ANSWER TO 3 DECIMAL PLACES

3. Interpret your results from part 2. in context of heights of ten-year-old children.

1. Interpret your results from part 2 in context of heights of ten-year-old children.

Answer 1

2)

Here, μ = 53.1, σ = 5.8 and x = 52.5. We need to compute P(X <= 52.5). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z = (52.5 - 53.1)/5.8 = -0.1

Therefore,
P(X <= 52.5) = P(z <= (52.5 - 53.1)/5.8)
= P(z <= -0.1)
= 0.4602

Here, μ = 53.1, σ = 5.8 and x = 53.7. We need to compute P(X >= 53.7). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z = (53.7 - 53.1)/5.8 = 0.1

Therefore,
P(X >= 53.7) = P(z <= (53.7 - 53.1)/5.8)
= P(z >= 0.1)
= 1 - 0.5398 = 0.4602

3)

probability that a randomly chosen heights of 10 year child has a height of less than 52.5 inches is 0.4602

probability that a randomly chosen heights of 10 year child has a height of more than 53.7inches is 0.4602