Math Problem Statement

Solve this

Let X represent the full height of a certain species of tree. Assume that X has a normal probability distribution with mean 128.4 ft and standard deviation 92.3 ft. You intend to measure a random sample of n = 152 trees. The bell curve represents the distribution of these sample means. The scale on the horizontal axis is the standard error (standard deviation) of the sampling distribution. Complete the indicated boxes, correct to two decimal places.

Solution

To solve the problem shown in the image, let's break down the steps:

Given Information:

  • The random variable Xˉ\bar{X} represents the mean height of a species of tree.
  • μ=128.4\mu = 128.4 ft (population mean height).
  • σ=92.3\sigma = 92.3 ft (population standard deviation).
  • Sample size n=152n = 152 trees.

We are asked to complete the labels for the sampling distribution of the sample means.

Key Formulae:

  1. Mean of the sampling distribution ( μXˉ\mu_{\bar{X}} ): The mean of the sampling distribution of the sample mean is the same as the population mean: μXˉ=μ=128.4ft\mu_{\bar{X}} = \mu = 128.4 \, \text{ft}

  2. Standard Error of the Mean ( σXˉ\sigma_{\bar{X}} ): The standard error is the standard deviation of the sample mean, and it is given by: σXˉ=σn\sigma_{\bar{X}} = \frac{\sigma}{\sqrt{n}} Substituting the known values: σXˉ=92.3152=92.312.32887.49ft\sigma_{\bar{X}} = \frac{92.3}{\sqrt{152}} = \frac{92.3}{12.3288} \approx 7.49 \, \text{ft}

Solution:

  • μXˉ=128.4ft\mu_{\bar{X}} = 128.4 \, \text{ft}
  • σXˉ=7.49ft\sigma_{\bar{X}} = 7.49 \, \text{ft}

Thus, the labels in the indicated boxes should be:

  • Mean of the sample means ( μXˉ\mu_{\bar{X}} ): 128.4
  • Standard Error ( σXˉ\sigma_{\bar{X}} ): 7.49

Would you like more details on any part of this solution?

Here are 5 related questions:

  1. What is the Central Limit Theorem, and how does it apply here?
  2. Why do we use σ/n\sigma / \sqrt{n} to calculate the standard error?
  3. How does increasing the sample size nn affect the standard error?
  4. What is the difference between standard deviation and standard error?
  5. How would the answer change if the sample size were doubled?

Tip: The larger the sample size, the smaller the standard error becomes, which makes the sample mean more precise.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Statistics
Sampling Distribution
Normal Distribution

Formulas

Mean of the sampling distribution: μ_X̄ = μ
Standard error: σ_X̄ = σ / √n

Theorems

Central Limit Theorem

Suitable Grade Level

College Level (Introductory Statistics)