Math Problem Statement

An experiment involves randomly selecting a sample of 256 middle managers for study. One item of interest is their mean annual income. The sample mean is computed to be $35,420 and the sample standard deviation is $2,050. What is the standard error of the mean?

Solution

The standard error of the mean (SEM) is calculated using the formula:

SEM=sn\text{SEM} = \frac{s}{\sqrt{n}}

where:

  • ss is the sample standard deviation
  • nn is the sample size

In this problem:

  • s=2050s = 2050
  • n=256n = 256

Substitute these values into the formula:

SEM=2050256=205016=128.125\text{SEM} = \frac{2050}{\sqrt{256}} = \frac{2050}{16} = 128.125

Thus, the standard error of the mean is:

SEM=128.125\text{SEM} = 128.125

Would you like more details on this calculation or have any questions?


Related Questions:

  1. How would the standard error change if the sample size increased?
  2. Why is the standard error important in estimating the population mean?
  3. What role does sample size play in reducing the standard error?
  4. How would we interpret the standard error in the context of this experiment?
  5. What is the relationship between sample standard deviation and standard error?

Tip:

Increasing the sample size decreases the standard error, making the estimate of the population mean more precise.

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Math Problem Analysis

Mathematical Concepts

Statistics
Standard Error
Sampling

Formulas

Standard Error of the Mean (SEM) = s / √n

Theorems

Central Limit Theorem

Suitable Grade Level

Undergraduate Statistics