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 (μ\mu) = 18
  • Standard deviation (σ\sigma) = 6.4
  • Sample size (nn) = 11

Since this is a normal distribution, for any sample of size nn, the sampling distribution properties are as follows:

  1. Mean of the sampling distribution of Xˉ\bar{X}:

    μXˉ=μ\mu_{\bar{X}} = \mu

  2. Standard deviation of the sampling distribution of Xˉ\bar{X} (also called the standard error):

    σXˉ=σn\sigma_{\bar{X}} = \frac{\sigma}{\sqrt{n}}

Now let’s apply these formulas to find each answer:

(a) Mean of the Sampling Distribution

Since the mean of the sampling distribution μXˉ\mu_{\bar{X}} is the same as the population mean:

μXˉ=18\mu_{\bar{X}} = 18

(b) Standard Deviation of the Sampling Distribution

The standard deviation of the sampling distribution σXˉ\sigma_{\bar{X}} is:

σXˉ=6.411\sigma_{\bar{X}} = \frac{6.4}{\sqrt{11}}

Calculating σXˉ\sigma_{\bar{X}}:

σXˉ=6.4111.929\sigma_{\bar{X}} = \frac{6.4}{\sqrt{11}} \approx 1.929

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:

  1. How does sample size affect the standard deviation of the sampling distribution?
  2. Why does the mean of the sampling distribution equal the population mean?
  3. What would happen to the standard deviation if the sample size were doubled?
  4. How would the sampling distribution differ for a non-normal population?
  5. 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