Math Problem Statement

A simple random sample of 40 items resulted in a sample mean of 25. The population standard deviation is σ = 5.

What is the standard error of the mean, σxbar?

Solution

The standard error of the mean (denoted as σxˉ\sigma_{\bar{x}}) is calculated using the formula:

σxˉ=σn\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}

Where:

  • σ\sigma is the population standard deviation (55),
  • nn is the sample size (4040).

Calculation:

σxˉ=540\sigma_{\bar{x}} = \frac{5}{\sqrt{40}}

First, calculate 40\sqrt{40}: 406.3246\sqrt{40} \approx 6.3246

Now divide: σxˉ=56.32460.7906\sigma_{\bar{x}} = \frac{5}{6.3246} \approx 0.7906

Final Answer:

The standard error of the mean is approximately 0.791.

Would you like more explanation or have any further questions?


Related Questions:

  1. How is the standard error of the mean different from the population standard deviation?
  2. What happens to the standard error if the sample size increases?
  3. How would the result change if the population standard deviation were unknown?
  4. Why is it important to use the sample size in calculating the standard error of the mean?
  5. Can the standard error be used to construct a confidence interval?

Tip:

When calculating the standard error, always verify whether the standard deviation provided is for the population (σ\sigma) or the sample (ss), as they involve different treatments in the formulas.

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Math Problem Analysis

Mathematical Concepts

Statistics
Standard Error
Probability

Formulas

σx̄ = σ / √n

Theorems

Central Limit Theorem

Suitable Grade Level

Grades 10-12