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A sociologist studies the income in a large city. The distribution of salaries is usually not normal (it is skewed).
Let \(X_i\) be the annual salary of a single person. Statistics show that the population mean is \(\mu = \dollar 45,000\) and the standard deviation is \(\sigma = \dollar 12,000\).
The sociologist surveys 100 random people. Let \(\overline{X}_{100}\) be the average salary of this sample.
  1. Find the expected average salary of the sample, \(E(\overline{X}_{100})\).
  2. Calculate the standard deviation for a single salary (\(\sigma\)), and then find the standard deviation of the sample mean salary, \(\sigma(\overline{X}_{100})\).
  3. Can we assume the average salary of the sample \(\overline{X}_{100}\) is normally distributed? Explain.
  4. Find the probability that the average salary of the sample exceeds \(\dollar 48,000\).

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