Many stock market observers say that when the P$/$E ratio for stocks gets over $20,$ the market is overvalued. The P$/$E ratio is the stock price divided by the most recent $12$ months of earnings. Suppose you are interested in seeing whether the current market is overvalued and would also like to know what proportion of companies pay dividends. A random sample of $30$ companies listed on the New York Stock Exchange is provided. $($Barron$$s$,$ January $19,2004)$

a$.$ What is a point estimate of the P$/$E ratio for the population of stocks listed on the New York Stock Exchange? Also, develop a $95\%$ confidence interval.

b$.$ Based on your answer to part $($a$),$ do you believe that the market is overvalued?

c$.$ What is a point estimate and $95\%$ confidence interval of the proportion of companies on the NYSE that pay dividends? Is the sample size large enough to justify using the normal distribution to construct a confidence interval for this proportion? Explain.