Math Problem Statement
Solve this
Solution
To solve the problem shown in the image, let's break down the steps:
Given Information:
- The random variable represents the mean height of a species of tree.
- ft (population mean height).
- ft (population standard deviation).
- Sample size trees.
We are asked to complete the labels for the sampling distribution of the sample means.
Key Formulae:
-
Mean of the sampling distribution ( ): The mean of the sampling distribution of the sample mean is the same as the population mean:
-
Standard Error of the Mean ( ): The standard error is the standard deviation of the sample mean, and it is given by: Substituting the known values:
Solution:
Thus, the labels in the indicated boxes should be:
- Mean of the sample means ( ): 128.4
- Standard Error ( ): 7.49
Would you like more details on any part of this solution?
Here are 5 related questions:
- What is the Central Limit Theorem, and how does it apply here?
- Why do we use to calculate the standard error?
- How does increasing the sample size affect the standard error?
- What is the difference between standard deviation and standard error?
- 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.
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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)
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