1. What do the following stand for?
a. ^{4}Σ_{i=1} x^{i-1}
b. ^{4}Σ_{i=1} (i+4)
^{c. 3}Σ_{i=1}^{2}Σ_{j=1} x_{i}y_{j}
2. The annual unemployment rate in a certain locality is determined by averaging the monthly unemployment rates for the year. Let the monthly unemployment rates be X1, X2,…X12 so that the annual rate is
X‾=1/12 ^{12}Σ_{i=1} X_{i}
Suppose now that we are given the average for the first eleven months,
X‾_{11}= 1/11 ^{11}Σ_{i=1} X_{i}
What would the value of X_{12} have to be if
(i) X‾ = X‾_{11}
(ii) X‾ =11/12 X‾_{11}
(iii) X‾ = X_{12}
3. Consider three fair tosses of a coin, and let X = number of heads and Y = number of changes in the sequence of toss results (e.g., HHH has no change of sequence, HTH has two changes of sequence, etc.)
a. Construct the sample space of all outcomes of this experiment and tabulate the marginal probability distributions of X and Y.
b. Tabulate the joint probability distribution of X and Y in the form of a two-way table.
c. Tabulate the conditional probability distribution of X given Y = 1.
d. Find the values of E(X), Var(Y), E(Y), Var(Y), and Cov(X,Y).
e. Find the values of E(X|Y=1) and Var(X|Y=1).
f. Find the value of Var(2X-3Y).
g. Are X and Y uncorrelated? Are X and Y independent? describe your answers.
4. Mr. Slick makes the following offer to you. You can toss four coins (furnished by him), and he will pay you an amount equal to the square of the number of heads showing (e.g., if you get three heads, he will pay you $9). In order to play this game you must pay $4 for each four-coin toss. On the naive assumption that the coins are fair, what is your expected gain?
5. Let X be a normally distributed variable with mean μ and variance σ
2. A random sample of three observations was obtained from this population. Consider the following estimators of μ.
μ^{^}_{1 }= X_{1}+X_{2}+X_{3}/3 μ^{^}_{2} = X_{1}/6+X_{2}/3+ X_{3}/2
a. Derive the expected value and variance of each estimator.
b. Compare the properties of these estimators in terms of unbiasedness, efficiency, and consistency.
6. You are told that X ~ N(8,36). Based on a sample of 25 observations, it was found that X‾= 7.5
a. What is the sampling distribution of X‾?
b. What is the probability of obtaining an X‾ = 7.5 or less?
c. Construct a 95% confidence interval for the population mean.
7. The following random sample was obtained from a normal population with mean μ and variance σ^{2}:
8,9,6,13,11,8,12,5,4,14.
a. find out the sample mean X‾ and the sample variance S^{2} and construct a 95% confidence interval for the population mean.
b. Test:μ= 7 against μ ≠7.
c. Test:μ = 7 against μ > 7.
Note: use α = 5%.
d. Now assume that σ = 3. Redo part b and part c and find out the p value for each test.
8. You are the manager of some toll stations on a highway. To find out if your booth operators pocket any of the money they handle, you compare the revenue each operator generates with the average of all operators. You fire any attendant who consistently generates less revenue than the average. Suppose the distribution of tolls collected per day per booth is normal with a mean of $280 per day and a standard deviation of $35. You've hired a new person and recorded the following daily revenues for his toll station: $225, $235, $275, $360, $359, $175, and $150. If you are willing to accept a 5% chance of mistakenly firing someone who is honest, should you fire him? What would be the P value for your test?