(a) \( (7\%) \) Find \( E(Y) \), the mean of \( Y \).

(b) \( (8\%) \) Apply the kurtosis of a standard normal distribution is 3 to find \( Var(Y) \), the variance of \( Y \).

(a) \( (8\%) \) Find \( f_{Y}(y) \), the marginal density of \( Y \).

(b) \( (7\%) \) Find \( f_{X / Y}(x|Y = \frac{1}{2}) \), the conditional density of \( X \) given \( Y = \frac{1}{2} \).

3. (10%) Suppose that using a rapid influenza diagnostic test (RIDT) the probability of a false positive is 0.1 and the probability of a false negative is 0.2. According to the historical data, a person suggested by a doctor to use the rapid test have a 60% chance of contracting an influenza. If a person suggested by a doctor to use the rapid test and the test result is negative, find the probability that this person contracting an influenza.[Note that a false positive is a test result that indicates a contracting when one does not; a false negative is a test result that indicates no contracting while in fact one does.]

4. (10%) A poll of President election was held. A random sample of 600 citizens were surveyed and there were 240 citizens would vote for candidate H. Find a 95% confidence interval for the percentage of ballots (turnout rate) of candidate H.

Part II The significance level for hypothesis testing is 0.05, unless otherwise specified.(50%)1. (10%) A study was instituted to learn how the diets of women changed during the winter and the summer. A random group of 12 women were observed during the month of July, and the percentage of each woman's calories that came from fat was determined. Similar observations were made on a different randomly selected group of size 12 during the month of January. Suppose the results were as follows:July: mean =30.83, standard deviation=4.30January: mean =35.13, standard deviation=4.51If we assume the population standard deviations for all months are equal, conduct a test for testing the hypothesis if the mean fat intake is the same for both months.

(a) What is the mean square within?

(b) What is the test statistics for testing the null hypothesis that the mean sales are equal for three layouts?

(a) What is the point estimate of the difference between two population proportions in favor of tax cut?

(b) Test the hypothesis that two population proportions in favor of tax cut are the same.

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