The mean height of fully grown birch trees are 18 feet with a standard deviation of 1.2 feet, normally distributed. Random samples of size 9 are drawn from this population and the mean of each sample is determined. Find that mean and the standard error of the mean, and the standard deviation of the sampling distribution of the sample means.

To find the mean and standard error of the mean for random samples of size 9 drawn from a population with a mean height of 18 feet and a standard deviation of 1.2 feet, we can use the following formulas:

1. Mean of the sample means (mean of the sampling distribution):
- The mean of the sample means is equal to the population mean.
- Therefore, the mean of the sample means is 18 feet.

2. Standard error of the mean:
- The standard error of the mean (SE) can be calculated using the formula:
SE = population standard deviation / √sample size.
- In this case, the population standard deviation is 1.2 feet and the sample size is 9.
- Calculating the standard error of the mean:
SE = 1.2 / √9 = 0.4 feet.

3. Standard deviation of the sampling distribution of the sample means:
- The standard deviation of the sampling distribution of the sample means (also known as the standard deviation of the sampling distribution, or simply, the standard deviation of the sample mean) can be calculated using the formula:
σ = population standard deviation / √sample size.
- In this case, the population standard deviation is 1.2 feet and the sample size is 9.
- Calculating the standard deviation of the sampling distribution of the sample means:
σ = 1.2 / √9 = 0.4 feet.

Therefore, the mean of the sample means is 18 feet, the standard error of the mean is 0.4 feet, and the standard deviation of the sampling distribution of the sample means is also 0.4 feet.

To find the mean and standard error of the mean, you can use the formula:

Mean = Population mean = 18 feet

Standard Error of the Mean = Population standard deviation / √(Sample size)
Standard Error of the Mean = 1.2 feet / √(9) = 1.2 feet / 3 = 0.4 feet

To find the standard deviation of the sampling distribution of the sample means, you can use the formula:

Standard Deviation of the Sampling Distribution of Sample Means = Population standard deviation / √(Sample size)
Standard Deviation of the Sampling Distribution of Sample Means = 1.2 feet / √(9) = 1.2 feet / 3 = 0.4 feet

The mean of each sample is the same as the population mean, which is 18 feet. The standard error of the mean and the standard deviation of the sampling distribution of the sample means are both 0.4 feet.