An alumni association at a university wants to gather data on the annual earning

Question

An alumni association at a university wants to gather data on the annual earnings of former students employed as accountants who earned their CPA licenses one year ago. After taking a random sample from university records of all accounting students who graduated a year ago, the alumni association sent out a survey to the selected individuals in the sample and received 75 responses. An analyst for the alumni association used the responses to calculate a 95% confidence interval for the mean salary of accountants one year after obtaining their CPA licenses, finding the interval to be $64,000 to $89,000 per year. Which of the following statements is the correct interpretation of this interval in context? Read carefully! The alumni association can be 95% confident, based on the method used to calculate the interval, that the true mean salary of the university's accounting graduates one year after obtaining their CPA licenses is between $64,000 and $89,000 per year. The alumni association can be 95% confident, based on the method used to calculate the interval, that any one of the university's accounting graduates one year after obtaining a CPA license will earn between $64,000 and $89,000 per year. The alumni association can be confident, based on the method used to calculate the interval, that the mean salary of all university students in the U.S. who graduated one year ago is $64,000 and $89,000 per year. The alumni association can be 95% confident, based on the method used to calculate the interval, that a large majority of the university's accounting graduates one year after obtaining their CPA license earn between $64,000 to $89,000 per year, based on the method used to construct the interval. There is a 95% chance, based on the method used to calculate the interval, that this particular interval, $64,000 to $89,000 per year, contains the true mean salary of the university's accounting graduates one year after obtaining their CPA licenses. The alumni association can be confident, based on the method used to calculate the interval, that 95% of the accountants in the university's sample of 75 earned between $64,000 and $89,000 per year.

Explanation / Answer

Correct interpretation:

"There is a 95% chance, based on the method used to calculate the interval, that this particular interval, $64,000 to $89,000 per year, contains the true mean salary of the university's accounting graduates one year after obtaining their CPA licenses".

Explanation:

If we collect the entire data of university former students of accounting and all their yearly salaries one year after obtaining CPA their licences instead of just taking a sample, and find the mean salary of this entire data, that is, we find true mean or the population mean. And this population mean is a single value. Let's say, for example, the population mean(or true mean) is $70,000.

And since, it is not possible to calculate and arrive at this $70,000 directly due to time, cost and any other constraints or difficulties, we take a sample from the population and find a range of values for the population mean. But we know that the population mean is a single value and it does not change. Thus, we should say that the range contains the true mean but not that the true mean is between $64,000 and $89,000.

Confidence level of 95% is the probability or chance that the range contains the true mean which implies that there is 5% chance that the range does not contain the true mean, that is, true mean may be either more than $89,000 or less than $64,000. So, we are taking a risk of 5%. And thus we should not say 95% of the accountants earned between $64,000 and $89,000. This is a wrong interpretation. It is only our confidence level to ensure that the range we calculated contains the true mean salary.

The interpretation that said that any one of the university students earns between $64,000 and $89,000 is incorrect because the range is for true mean salary only but not for any particular individual salary. Individual salary may be extremely more or less but we are cocerned with the mean salary that is obtained from adding all salaries and dividing the sum by total number of students concerned. One, few or even many of the students may earn between those amounts but it does not become a correct interpretation.

Similarly, the interpretations saying all students and large majorityof students is also incorrect for the same reason above and there is also 5% chance that even the range does not contain the true mean salary.

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