The data set "Computer_Exercise_4.xlsx" is retrieved from I/B/E/S International Inc. It contains buy-sell-hold recommendations for the stock of Apple INC (Ticker code "AAPL") from a wide cross section of individual professional analyst for the time period between September 2004 and September 2014.

1. By definition, a random variable, ????, needs to have a numerical outcome. Generate a new series of outcomes, ????, corresponding the analysts' recommendations as follows:

????????=0.5 if recommendation is "STRONG BUY"

????????=0.4 if recommendation is "BUY"

????????=0.3 if recommendation is "HOLD"

????????=0.2 if recommendation is "UNDERPERFORM"

????????=0.1 if recommendation is "SELL"

2. Assume that the analysts' recommendations in the data set are a random sample drawn from an underlying population of analysts' recommendation, ????. ???? has a p.d.f. with unknown mean, , and unknown variance, 2.

Compute an unbiased estimate for and an unbiased estimate for 2.

3. Denote the sample-average estimate as

????=1????????????????????=1

What is the standard error of ????, i.e. ????????(????)?

4. Clearly the population, ????, cannot be normally distributed (???? only has 5 possible discrete outcomes; a normally distributed variable is continuous). Assuming that ???? (i.e. the number of observations in your sample) is large enough, compute the following approximate asymptotic confidence intervals for : 90%-, 95%-, and 99%-confidence interval.

5. Interpret the confidence intervals in 4. What can you conclude with respect to the (unknown) true population mean of analysts' recommendation?

6. Compute the 95%-confidence interval for , again, this time relying on the rule of thumb (see lecture slides). Is this interval very different from the 95%-confidence interval computed in 4.?

7. Under the same assumptions as 4. (i.e. ???? is large enough), evaluate the null hypothesis that the average recommendation in the population is "BUY", at a significance level of 10%,

a) Against the alternative hypothesis that the average recommendation is stronger (more favorable) than "BUY".

b) Against the alternative hypothesis that the average recommendation is weaker (less favorable) than "BUY".

c) Against the alternative hypothesis that the average recommendation is not equal to "BUY".

Rely on the standard approach to hypothesis testing. That is, compare the outcome of your test statistic to the respective critical value.

8. Perform the same hypotheses tests as in 7., but now rely on the ????-values. Comment on the outcomes of the tests.