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
Where:
X = the height we want to find the probability for (either 32.8 or 36.4)
μ = the mean height (35.2 inches)
σ = the standard deviation (1.2 inches)
For 32.8 inches:
z = (32.8 - 35.2) / 1.2 = -2
For 36.4 inches:
z = (36.4 - 35.2) / 1.2 = 1
We can now use a standard normal distribution table or a calculator to find the probability that a randomly selected 3-year-old is between these two heights. The probability is:
P(-2 < z < 1) ≈ 0.8186
Therefore, the probability that a randomly selected 3-year-old is between 32.8 inches and 36.4 inches in height is approximately 0.8186 or 81.86%.
B) If a 3-year-old is 37 inches tall, we can use the same z-score formula to find the z-score:
z = (37 - 35.2) / 1.2 ≈ 1.5
By looking at the standard normal distribution table or using a calculator, we see that a z-score of 1.5 corresponds to a probability of approximately 0.9332. This means that only about 93.32% of 3-year-olds are below 37 inches in height, making a 3-year-old who is 37 inches tall above average in height for their age group.