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 calculate the probability of a randomly selected 3-year-old being between 32.8 inches and 36.4 inches in height, we first need to standardize these values using the z-score formula:
Z = (X - μ) / σ
For 32.8 inches:
Z1 = (32.8 - 35.2) / 1.2 = -2.00
For 36.4 inches:
Z2 = (36.4 - 35.2) / 1.2 = 1.00
Next, we look up these z-scores in a standard normal distribution table to find the corresponding probabilities:
P(Z < -2.00) = 0.0228
P(Z < 1.00) = 0.8413
Since we are interested in the probability between these two values, we subtract the smaller probability from the larger one:
P(32.8 < X < 36.4) = P(Z < 1.00) - P(Z < -2.00) = 0.8413 - 0.0228 = 0.8185
Therefore, 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.) A 3-year-old who is 37 inches tall is beyond one standard deviation above the mean. Since the mean height for 3-year-olds is 35.2 inches with a standard deviation of 1.2 inches, being 37 inches tall puts this child at a height of 1.5 standard deviations above the mean.
According to the empirical rule for normal distributions, approximately 68% of the data falls within one standard deviation of the mean and about 95% falls within two standard deviations. This means that approximately 95% of 3-year-olds fall within 2 standard deviations of the mean, and being 1.5 standard deviations above the mean is statistically above average for this age group.
Therefore, a 3-year-old who is 37 inches tall would be considered above average in height for their age group.