Problem: In this exercise we consider data of 474 employees (working in the banking sector) on the variables y (the yearly salary in dollars) and x (the number of finished years of education).

1. Make histograms of the variables x and y and make a scatter plot of y against x.
2. Compute mean, median, and standard deviation of the variables x and y and compute the correlation between x and y. Check that the distribution of the salaries y is very skewed and has excess kurtosis.
3. Compute a 95% interval estimate of the mean of the variable y, assuming that the salaries are NID( \(\mu \), \({{\sigma }^{2}}\) ).
4. Define the random variable z = log (y). Make a histogram of the resulting 474 values of z and compute the mean, median, standard deviation, skewness and kurtosis of z. Check that \(\bar{z}\ne \log \left( {\bar{y}} \right)\) but that med(z) = log (med(Y) ). Explain this last result.
5. Compute a 95% interval estimate of the mean of the variable z, assuming that the observations on z are NID( \({{\mu }_{Z}}\), \(\sigma _{Z}^{2}\) )
6. If z \(\tilde{\ }\) NID( \({{\mu }_{Z}}\), \(\sigma _{Z}^{2}\) ), then y = e is said to be log- normally distributed. Show that the mean of y is given by p \(\mu ={{e}^{{{\mu }_{Z}}+\frac{1}{2}\sigma _{Z}^{2}}}\).
7. Compute a 95% interval estimate of p, based on the results in d, e, and f. Compare this interval with that obtained in c. Which interval do you prefer?

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Solution: The downloadable solution consists of 8 pages, 647 words and 8 charts.
Deliverable: Word Document