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How do I identify standard error of the mean and determine probabilities of a sample mean?

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In the early years of the twenty-first century, annual salaries for federal employees in the Upper Midwest averaged approximately $42,000, with a population standard deviation of $4,900. If we drew a random sample of 25 federal employees from this population:

What would the standard error of the mean be?
What is the probability that the sample mean would be between $41,000 and $43,000?
What is the probability that the sample mean would be between $41,000 and $43,000 if we increased the sample size to 100?

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Solution Summary

The standard errors of the mean and determine probabilities are determined. Probability sample means are examined.

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In the early years of the twenty-first century, annual salaries for federal employees in the Upper Midwest averaged approximately $42,000, with a population standard deviation (?) of $4,900. If we drew a random sample of 25 federal employees from this population:

1) What would the standard error of the mean be?

Given that sigma=$4,900 and n=25. The standard error of the mean would be

...

Solution provided by:
Changping Wang, MA
Education
  • BSc , Wuhan Univ. China
  • MA, Shandong Univ.
Recent Feedback
  • "Your solution, looks excellent. I recognize things from previous chapters. I have seen the standard deviation formula you used to get 5.154. I do understand the Central Limit Theorem needs the sample size (n) to be greater than 30, we have 100. I do understand the sample mean(s) of the population will follow a normal distribution, and that CLT states the sample mean of population is the population (mean), we have 143.74. But when and WHY do we use the standard deviation formula where you got 5.154. WHEN & Why use standard deviation of the sample mean. I don't understand, why don't we simply use the "100" I understand that standard deviation is the square root of variance. I do understand that the variance is the square of the differences of each sample data value minus the mean. But somehow, why not use 100, why use standard deviation of sample mean? Please help explain."
  • "excellent work"
  • "Thank you so much for all of your help!!! I will be posting another assignment. Please let me know (once posted), if the credits I'm offering is enough or you ! Thanks again!"
  • "Thank you"
  • "Thank you very much for your valuable time and assistance!"
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