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 54.8 inches, and standard deviation of 6.5 inches.
A) What is the probability that a randomly chosen child has a height of less than 53.75 inches?
Answer= (Round your answer to 3 decimal places.)
B) What is the probability that a randomly chosen child has a height of more than 57.9 inches?
Answer= (Round your answer to 3 decimal places.)
1. **Problem Statement:**
We have a normal distribution for heights of ten-year-old children with mean $\mu = 54.8$ inches and standard deviation $\sigma = 6.5$ inches.
We want to find:
A) $P(X < 53.75)$
B) $P(X > 57.9)$
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2. **Formula and Important Rules:**
For a normal distribution, to find probabilities, we convert the value $X$ to a standard normal variable $Z$ using:
$$Z = \frac{X - \mu}{\sigma}$$
Then use standard normal distribution tables or a calculator to find probabilities.
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3. **Part A: Probability that height is less than 53.75 inches**
Calculate $Z$:
$$Z = \frac{53.75 - 54.8}{6.5} = \frac{-1.05}{6.5} = -0.1615$$
Using standard normal tables or calculator, find $P(Z < -0.1615)$.
From tables or calculator, $P(Z < -0.1615) \approx 0.436$ (rounded to 3 decimals).
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4. **Part B: Probability that height is more than 57.9 inches**
Calculate $Z$:
$$Z = \frac{57.9 - 54.8}{6.5} = \frac{3.1}{6.5} = 0.4769$$
Find $P(Z > 0.4769)$.
Using tables or calculator, $P(Z > 0.4769) = 1 - P(Z < 0.4769) \approx 1 - 0.683 = 0.317$ (rounded to 3 decimals).
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**Final answers:**
A) $0.436$
B) $0.317$