The height of a certain species of a tree are normally distributed with a mean of 30ft and stand dev of 4ft. If a random sample of 16 of these trees is taken, what is the probability that the sample mean is less than 32fy?
z = (32-30)/4sqrt(16))
z = 2.
Z table
To solve this problem, we will use the concept of sampling distributions and the Central Limit Theorem.
Step 1: Determine the distribution of the sample mean
Given that the heights of the trees are normally distributed, the sample mean of heights will also be normally distributed. However, the mean and standard deviation of the sample mean will differ from the population parameters.
The mean of the sample mean (∑x/n) will be the same as the population mean, which is 30ft.
The standard deviation of the sample mean (σ/√n) will be the population standard deviation divided by the square root of the sample size. In our case, the population standard deviation is 4ft and the sample size is 16. So, the standard deviation of the sample mean is 4ft/√16 = 4ft/4 = 1ft.
Step 2: Standardize the sample mean using z-score
To find the probability that the sample mean is less than 32ft, we need to standardize this value using the z-score formula:
z = (x - μ) / (σ/√n)
where x is the value we want to standardize, μ is the population mean, σ is the population standard deviation, and n is the sample size.
In this case, we want to standardize x = 32ft, μ = 30ft, σ = 4ft, and n = 16.
z = (32 - 30) / (4/√16) = 2 / (4/4) = 2.
Step 3: Find the probability using the standard normal distribution table
Now that we have the z-score, we can use the standard normal distribution table (also known as the z-table) to find the probability associated with that z-score.
The z-table provides the probability of a standard normal distribution being less than a given z-value. In this case, we are interested in finding the probability that the sample mean is less than 32ft, which corresponds to a z-score of 2.
Using the z-table, you can find that the area to the left of a z-score of 2 is approximately 0.9772.
Therefore, the probability that the sample mean is less than 32ft is approximately 0.9772 or 97.72%.
Note: If you have access to statistical software or calculators, you can also use them to find the probability directly without using the z-table.