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 standardize the values using z-scores.
First, find the z-scores:
Z1 = (32.8 - 35.2) / 1.2 = -2
Z2 = (36.4 - 35.2) / 1.2 = 1
Next, use a standard normal distribution table or a calculator to find the area between these two z-scores:
P(-2 < Z < 1) = P(Z < 1) - P(Z < -2)
From the standard normal distribution table, we find that P(Z < 1) is approximately 0.8413 and P(Z < -2) is approximately 0.0228.
Therefore, P(-2 < Z < 1) = 0.8413 - 0.0228 = 0.8185
So, the probability that a randomly selected 3-year-old is between 32.8 inches and 36.4 inches in height is approximately 0.8185, or 81.85%.
b.) Since the average height of a 3-year-old is 35.2 inches, a 3-year-old who is 37 inches tall would be considered above average in height for their age group. This is because 37 inches is greater than the average height of 35.2 inches, indicating that the child is taller than most 3-year-olds in the preschool population.