https://file.aiquickdraw.com/mathbot/file/1729026961920yt0ey7ht.jpg

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

### Given Information:
- The random variable \( \bar{X} \) represents the mean height of a species of tree.
- \( \mu = 128.4 \) ft (population mean height).
- \( \sigma = 92.3 \) ft (population standard deviation).
- Sample size \( n = 152 \) trees.

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

### Key Formulae:

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

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

### Solution:
- \( \mu_{\bar{X}} = 128.4 \, \text{ft} \)
- \( \sigma_{\bar{X}} = 7.49 \, \text{ft} \)

Thus, the labels in the indicated boxes should be:
- **Mean of the sample means ( \( \mu_{\bar{X}} \) ):** 128.4
- **Standard Error ( \( \sigma_{\bar{X}} \) ):** 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 \( \sigma / \sqrt{n} \) to calculate the standard error?
3. How does increasing the sample size \( n \) 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.

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.

Solve a statistics problem involving the sampling distribution of tree heights with a mean of 128.4 ft and a standard deviation of 92.3 ft. Calculate the standard error for a sample size of 152 trees, and understand how the Central Limit Theorem applies to this problem.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation