The heights of 3-year-olds at a preschool averaged 35.2 inches with a standard deviation of 1.2 inches. a.) Assuming a normal distribution, estimate the probability that a randomly selected 3-year-old is between 32.8 inches and 36.4 inches in height. b.) If a 3-year-old is 37 inches tall, are they considered to be above average in height for their age group? Explain your answer.
a.) To estimate the probability that a randomly selected 3-year-old is between 32.8 inches and 36.4 inches in height, we can use the z-score formula:
z = (X - μ) / σ
For 32.8 inches:
z = (32.8 - 35.2) / 1.2 = -2.0
For 36.4 inches:
z = (36.4 - 35.2) / 1.2 = 1.0
Using a standard normal distribution table, we find the following probabilities:
P(z < -2.0) = 0.0228
P(z < 1.0) = 0.8413
Therefore, the probability that a randomly selected 3-year-old is between 32.8 inches and 36.4 inches in height is:
P(-2.0 < z < 1.0) = P(z < 1.0) - P(z < -2.0) = 0.8413 - 0.0228 = 0.8185
b.) If a 3-year-old is 37 inches tall, we need to calculate the z-score as follows:
z = (37 - 35.2) / 1.2 = 1.5
Using the standard normal distribution table, we find that P(z < 1.5) = 0.9332. This means that a 3-year-old who is 37 inches tall is above approximately 93.32% of their peers in terms of height. Therefore, they would be considered above average in height for their age group.