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Conducting a hypothesis test for difference in means using a z-statistic

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A study was conducted on the annual incomes of corporate trainers in the state of New York, in metropolitan areas of less than 100,000, and in metropolitan areas having a population of over 500,000. Some sample statistics are:

Sample Statistic Population less than 100K Population more than 500K
Sample Size 45 60
Sample Mean \$31,290 \$31,330
Sample SD \$1,060 \$1,900

Test the hypothesis that the annual income of corporate trainers in areas of more then 500,000 are significantly more than those in areas of less than 100,000. Use the 5% level of risk.

Solution Summary

The following outlines the process of conducting a hypothesis test to determine if there is a significant difference in means for two samples. Formulas and work are provided in a Word document.

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A study was conducted on the annual incomes of corporate trainers in the state of New York, in metropolitan areas of less than 100,000, and in metropolitan areas having a population of over 500,000. Some sample statistics are:
Sample Statistic Population less than 100K Population more than 500K
Sample Size 45 60
Sample Mean \$31,290 ...

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• 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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