Math Problem Statement
A measurement is normally distributed with mean=18 and slope=6.4
. Round answers below to three decimal places.
(a) The mean of the sampling distribution of for samples of size 11 is:
(b) The standard deviation of the sampling distribution of for samples of size 11 is:
Solution
To solve this, let's go over each part. Here’s the approach for each:
Given:
- Mean () = 18
- Standard deviation () = 6.4
- Sample size () = 11
Since this is a normal distribution, for any sample of size , the sampling distribution properties are as follows:
-
Mean of the sampling distribution of :
-
Standard deviation of the sampling distribution of (also called the standard error):
Now let’s apply these formulas to find each answer:
(a) Mean of the Sampling Distribution
Since the mean of the sampling distribution is the same as the population mean:
(b) Standard Deviation of the Sampling Distribution
The standard deviation of the sampling distribution is:
Calculating :
Final Answers:
- (a) The mean of the sampling distribution is 18.000.
- (b) The standard deviation of the sampling distribution is 1.929.
Would you like additional clarification on any part of the solution?
Related Questions:
- How does sample size affect the standard deviation of the sampling distribution?
- Why does the mean of the sampling distribution equal the population mean?
- What would happen to the standard deviation if the sample size were doubled?
- How would the sampling distribution differ for a non-normal population?
- Can the central limit theorem be applied here with a sample size of 11?
Tip:
When calculating the standard deviation of a sampling distribution, remember that increasing the sample size reduces the standard error, making the sample mean more precise.
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Math Problem Analysis
Mathematical Concepts
Statistics
Normal Distribution
Sampling Distribution
Formulas
Mean of the sampling distribution: μ_X̄ = μ
Standard deviation of the sampling distribution (Standard Error): σ_X̄ = σ / √n
Theorems
Central Limit Theorem
Suitable Grade Level
Grades 10-12
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