Question

1. the age at which children begin to walk independently follows an approximately normal distribution with mean 12 months and standard deviation 1.5 months. children who walk before 10 months of age are considered \early walkers.\ given that a randomly selected child is in the lowest 25 percent in age of beginning to walk, what is the probability that the child is considered an \early walker\? a) 0.091 b) 0.159 c) 0.365 d) 0.728 e) 0.909

Explanation:

Step1: Define normal distribution

Let $X$ = age to walk, $X \sim N(\mu=12, \sigma=1.5)$

Step2: Find 25th percentile age

Find $x_p$ where $P(X < x_p)=0.25$. Use z-score for 0.25: $z_p \approx -0.674$
$x_p = \mu + z_p\sigma = 12 + (-0.674)(1.5) = 12 - 1.011 = 10.989$ months

Step3: Find P(X < 10)

Calculate z-score for 10: $z = \frac{10 - 12}{1.5} = \frac{-2}{1.5} \approx -1.333$
$P(X < 10) = P(Z < -1.333) \approx 0.091$

Step4: Apply conditional probability

We need $P(X < 10 | X < x_p) = \frac{P(X < 10)}{P(X < x_p)}$
$\frac{0.091}{0.25} = 0.364 \approx 0.365$

Answer:

C) 0.365