Respuesta :
Given Information:
Mean height of ten year children = μ = 54.4 inches
Standard deviation = σ = 7.9 inches
Required Information:
A) What is the probability that a randomly chosen child has a height of less than 46.35 inches?
B) What is the probability that a randomly chosen child has a height of more than 38.6 inches?
Answer:
[tex]P(X < 46.35) = 15.38\%[/tex]
[tex]P(X > 38.6) = 97.73\%[/tex]
Explanation:
Let X is the random variable that represents height measurements of ten-year-old children with mean of 54.4 inches and standard deviation of 7.9 inches.
A) The probability that a randomly chosen child has a height of less than 46.35 inches is given by
[tex]P(X < 46.35) = P(Z < \frac{x - \mu}{\sigma} )[/tex]
[tex]P(X < 46.35) = P(Z < \frac{46.35 - 54.4}{7.9} )[/tex]
[tex]P(X < 46.35) = P(Z < -1.02)[/tex]
From the z-table , the z-score corresponding to -1.02 is 0.1538
[tex]P(X < 46.35) = 0.1538[/tex]
[tex]P(X < 46.35) = 15.38\%[/tex]
Therefore, 15.38% children has a height of less than 46.35 inches.
B) The probability that a randomly chosen child has a height of more than 38.6 inches is given by
[tex]P(X > 38.6) = 1 - P(Z < \frac{x - \mu}{\sigma} )[/tex]
[tex]P(X > 38.6) = 1 - P(Z < \frac{38.6 - 54.4}{7.9} )[/tex]
[tex]P(X > 38.6) = 1 -P(Z < -2)[/tex]
From the z-table , the z-score corresponding to -2 is 0.0227
[tex]P(X > 38.6) = 1 -0.0227[/tex]
[tex]P(X > 38.6) = 0.9773[/tex]
[tex]P(X > 38.6) = 97.73\%[/tex]
Therefore, 97.73% children has a height of greater than 38.6 inches.
