The heights of kindergarten children are normally distributed with a mean of 39.5 inches and a standard
deviation of 3.8 inches. In a kindergarten class of 23 children, what is the probability that their mean
height is between 38.1 and 41.2 inches?
Z = (score-mean)/SEm
SEm = SD/√n
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability between the two Z scores.
To solve this problem, we need to use the concept of sampling distribution. The sampling distribution of the mean is the distribution of sample means taken from all possible samples of the same size from a population. In this case, we know that the heights of kindergarten children are normally distributed with a mean of 39.5 inches and a standard deviation of 3.8 inches.
To find the probability that the mean height of a class of 23 children falls between 38.1 and 41.2 inches, we need to calculate the z-scores for these values and then use the standard normal distribution.
Step 1: Calculate the standard error of the mean (σ/√n):
The standard error of the mean is equal to the standard deviation of the population divided by the square root of the sample size.
Standard error (SE) = σ / √n
Standard error (SE) = 3.8 / √23
Standard error (SE) ≈ 3.8 / 4.7958
Standard error (SE) ≈ 0.7929
Step 2: Calculate the z-scores for the lower and upper limits:
The z-score is a measure of how many standard deviations an individual value is from the mean.
z = (x - μ) / σ
For the lower limit of 38.1 inches:
z1 = (38.1 - 39.5) / 0.7929
z1 ≈ -1.764
For the upper limit of 41.2 inches:
z2 = (41.2 - 39.5) / 0.7929
z2 ≈ 2.141
Step 3: Calculate the probability using the standard normal distribution:
Now that we have the z-scores for the lower and upper limits, we can find the probability associated with these z-scores using a standard normal distribution table or a calculator.
P(38.1 ≤ x ≤ 41.2) = P(-1.764 ≤ z ≤ 2.141)
Looking up these z-scores in a standard normal distribution table, we find that P(-1.764 ≤ z ≤ 2.141) is approximately 0.9251.
So, the probability that the mean height of a kindergarten class of 23 children falls between 38.1 and 41.2 inches is approximately 0.9251 or 92.51%.