The mean of each sample (regardless of the sample size) will also be 90 feet. So, the correct answer is C. 90 and 1.59.
The problem states that the heights of fully grown white oak trees are normally distributed with a mean of 90 feet and a standard deviation of 4.2 feet. This means the majority of trees will be around 90 feet tall, with some shorter and some taller, following a bell-shaped curve.
We are then interested in random samples of size 7, meaning we take 7 trees at random and calculate their average height. This average is called the sample mean.
Importantly, the mean of the sample means will tend towards the population mean (90 feet) as we take more and more samples. However, due to randomness, individual sample means will vary around the population mean.
The standard error of the mean (SEM) tells us how much we can expect the sample means to vary from the population mean. It is calculated as the standard deviation divided by the square root of the sample size:
SEM = σ / √n.
In this case,
SEM = 4.2 feet / √7 ≈ 1.59 feet.
Therefore, we can expect the mean of the sample means to be close to 90 feet, but with some variation due to randomness. The standard error of the mean tells us the typical range of this variation, which in this case is around 1.59 feet.
So, the correct answer is C. 90 and 1.59.
Question
Set up 2: The heights of fully grown white oak trees are normally distributed, with a mean of 90 feet and standard deviation of 4.2 feet. Random samples of size 7 are drawn from this population, and the mean of each sample is determined. The mean equals __ and standard error of the mean is ___.
A. 90 and 1.33
B. 4.2 and 90
C. 90 and 1.59
D. 1.33 and 90