To tackle this question, we need to understand the basics of normal distribution and how sample means behave in statistics. In our case, we have a population of fourth graders whose heights are normally distributed with a mean (μ) of 52 inches and a standard deviation (σ) of 3.5 inches.
When we take a simple random sample of size 30, we can utilize the Central Limit Theorem, which states that the sampling distribution of the sample mean will also be normally distributed, regardless of the shape of the population distribution, given the sample size is sufficiently large.
The mean of our sampling distribution (the mean of sample means) will still be 52 inches, while the standard deviation of the sampling distribution (also known as the standard error) can be calculated using the formula:
Standard Error (SE) = σ / √n
Where:
- σ = population standard deviation (3.5 inches)
- n = sample size (30)
Plugging in the numbers, we get:
SE = 3.5 / √30 ≈ 0.64 inches
Now that we have the mean and the standard error, we can find the probability that the average height (sample mean) of our sample of fourth graders exceeds a certain value. For instance, if we want to find the probability that the average height is above 53 inches, we need to standardize this value using a Z-score:
Z = (X – μ) / SE
Where:
- X = the threshold height (in this case, 53 inches)
- μ = mean of the sampling distribution (52 inches)
- SE = standard error (0.64 inches)
Substituting the values gives:
Z = (53 – 52) / 0.64 ≈ 1.56
Finally, we can use the Z-table or a calculator to find the probability corresponding to a Z-score of 1.56. This value tells us the area to the left of this Z-score. To find the complement (the area to the right), we subtract this probability from 1:
P(X > 53) = 1 – P(Z < 1.56)
This calculation will give you the probability that the average height of your sample of fourth graders is above 53 inches.