The National Science Foundation (NFS) condust its Survey of Earned Doctorates (SED) of all individuals receiving research doctoral degree from accredited U.S. institurions. The results from the 2010 survey published on the NSF website indicate that 78.2% of individuals earning their doctorate in the physical sciences have no graduate debt while 48.3% of those earning their docotorate in the social sciences have no graduate debt. Of the 48, 069 research doctorates granted in 2010, 93% completed the SED, thus the information collected by the NSF can be good approximations of the population parameters.
a) Suppose we take random samples of 100 individuals who earned a doctorate in the physical sciences in 2010 and 100 individuals who earned a docotrate in the social sciences in 2010. Find the mean and standard error (using four decimal places) of the distribution of differences in sample proportions and indicate if the sample sizes are large enough to use the Central Limit Theorem.
b) Now use random samples of 25 for those who earned a doctorate in physical sciences and 50 for those who earned a doctorate in social sciences. Find the mean and standar error and indicate if the sample sizes are large enough to use the Central Limit Theorem.
c) Now use random samples of 50 for the physical sciences and 25 for the social sciences and find the mean and standard error.