Question

In: Statistics and Probability

An agency wants to examine the distribution of wages F(x) of graduates with bachelor degrees. For...

An agency wants to examine the distribution of wages F(x) of graduates with bachelor degrees. For that purpose they receive from US Census Bureau database a sample X₁,…, X_n of 10,000 numeric records of wages for randomly chosen graduates.

Find the algorithm to provide an interval of wages which cover the 50% central part of distribution F(x) with approximate confidence 95%. How can you evaluate the accuracy of that estimation?

Expert Solution

Given the data, we calculate the following quantities:

a) The sample size

This is the size of the sample out of 10,000 records. Note that n should be at least 30 to use the normal approximation.

b) The sample mean

This corresponds to 50% central part of the distribution.

c) The sample standard deviation

As the sample size is large, we use the normal approximation for the distribution. We use a z-table for converting probability to the z-score.

d) The 95% confidence interval corresponds to a probability range of 0.025 to 0.975. Using a z-table, we get the corresponding z-score for 0.975 as 1.96.

e) In the final step, we get the confidence interval as:

All the quantities have been calculated in the previous steps.

To evaluate the estimation, we take more samples of size from the same data and confirm that at least 95% of those samples have their mean in the above confidence interval.

orchestra answered 1 year ago

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