Hello, if you have any need, please feel free to consult us, this is my wechat: wx91due
Econ-312: Assignment -1
1. Let X be a random variable with a discrete probability distribution with population mean μx . Consider the following relationships:
Variance:
Covariance:
(i) (5 points) Prove that the right hand side of A = right hand side of B. Show all the steps. (ii) (5 points) Prove that the rhs of C = rhs of D. Show all the steps.
2. Consider the following two random variables y1 and y2 , both are linearly related to another random variable x that has mean ( μx ) and variance ( σx(2) ), as follows:
y1 = a1 + b1x
y2 = a2 + b2x ,
where a1 , b1 , a2 , b2 are constants. For example vacation travel expenditure ( y1 ) may be linearly related to income ( x ) and the number of car ownership ( y2 ) may be linearly related to income ( x ). Based on this information, find the expressions for the following:
(i) (5 points) E(y1 ) =
(ii) (5 points) E(y2 ) =
(iii) (5 points) var(y1 ) =
(iv) (5 points) var(y2 ) =
(v) (5 points) covar(y1, y2 ) =
3. You all know that Unbiasedness and Efficiency are two most important properties of an estimator, which is also often called a sampling statistic. Consider the following two estimators:
where { X1 , X2 , X3 } is a sample of size n = 3 drawn independently from a population with mean (μx) and variance ( σx(2)).
(i) (5 points) Show that both estimators are unbiased.
(ii) (10 points) Prove that the estimator Y1 is more efficient relative to Y2 .
4. Using EXCEL, do the following in sequence:
(i) Generate two random samples ( x and y ) of size 10 from two Normal Distribution – one from N(μx = 50, σx(2) = 25) and the other from N(μy = 75, σy(2) = 64) . Arrange them in two columns one with column heading x and the other with column heading y .
(ii) Find in sequence the following summary statistics by showing the complete work in EXCEL:
a. (3 points) Sample mean: x =
b. (3 points) Sample mean: y =
c. (3 points) Sample variance: sx(2) =
d. (3 points) Sample variance: sy(2) =
e. (3 points) Sample correlation: rxy =
f. (5 points) Print out the EXCEL worksheet. I will not accept it if it is more than one page.
5. In 1960, census results indicated that the age at which American men first married had a mean of 23.3 years. It is widely suspected that young people today are waiting longer to get married. We want to find out if the mean age of first marriage has increased during the past 43 years.
a. (5 points) Write out the appropriate hypotheses.
b. (5 points) We plan to test our hypothesis by selecting a random sample of 64 men who married for the first time last year. Write down the approximate sampling distribution of the sample statistic X in your answer sheet by replacing the symbols below by numbers that you know from the information given.
X ~ Normal(µ, σ / 
n)
c. (5 points) The men in our sample married at an average age of 24.2 years with a standard deviation of 5.3 years. Carry out an appropriate test at 1% significance level, using the critical value approach. That is, find the appropriate critical region, make decision and write your conclusion.
d. (5 points) Find the p-value for the test and make sure that your acceptance-rejection decision remains the same if the p-value approach is adopted. Explain in your own words and as explained in class, how you interpret this p-value.
e. (5 points) Find a 99% confidence interval for the for the unknown population average age at marriage.
f. (5 points) Explain in your own words and as explained in class what you mean by this confidence interval.