ECON 222

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Home>Mathematics homework help>ECON 222

CONCORDIA UNIVERSITY

Department of Economics

ECON 222/4 SECTIONS A, B and BB

STATISTICAL METHODS II

WINTER 2016 – ASSIGNMENT 1

Due: Monday, February 15, before 3:00 pm

1. (4 marks) Simplify the following expressions.

a.

Find

b.

Find , where

2.

(4 marks) The population mean and variance of the random variable X are and , respectively. Prove that, for a sufficiently large sample size n and the sample mean

a.

b.

3. (6 marks) Let X be a random variable with a probability density function (PDF) given by

a. Solve for c.

b.

Calculate .

c.

Calculate .

4.

(2 marks) A random sample of n voters is selected to estimate the proportion of voters who plan to vote for Candidate A in an election, . How large does n need to be so that we can obtain a 90 percent confidence interval with a margin of error of .

5. (10 marks) Let X and Y be two continuous variables with a joint PDF given by

a. Calculate the marginal PDF of X.

b. Calculate the marginal PDF of Y.

c. Briefly explain if X and Y are independent.

d.

Calculate .

e.

Calculate .

6.

(10 marks) The data file assignment.xlsx contains the grades for 33 students on assignment 1 and assignment 2 . Let be normally and independently distributed with a mean and variance of and , respectively.

a.

Calculate .

b.

Calculate .

c. State the appropriate null and alternative hypotheses to test whether the performance on the assignments does not differ.

d. Briefly explain whether a t- or Z-test is more appropriate.

e. Perform the appropriate test at the 5-percent level of significance and briefly explain your conclusion.

7. (10 marks) The data file fultonfish.dat shows the daily sales of fish (in pounds) for a period of time.

a.

Test against at the 5-percent level of significance. Briefly explain your result using a diagram showing the estimated value of the test statistic and the critical value.

b. Calculate the p-value of the test statistic and briefly explain how it can be used to perform the hypothesis test. Show the p-value in the diagram.

Let total weekly sales be given by , where represents sales on weekday i.

c.

Calculate .

d.

Calculate .

e.

Derive the probability distribution of and calculate a 95-percent confidence interval estimate for .

8. (6 marks) A police chief claims that the standard deviation in the length of response times is less than 3.7 minutes. A random sample of 9 response times from a normal population has a standard deviation of 3.0 minutes.

a. State the appropriate null and alternative hypotheses.

b.

Briefly explain whether a t- or -statistic is more appropriate.

c.

Perform the appropriate test for and briefly explain your conclusion.

1

y = Ae a+bX( )

y=Ae

a+bX

()

µ

m

σ 2

s

2

x = 1 n

xi i=1

n

∑

x=

1

n

x

i

i=1

n

å

E x( ) = µ

Ex

()

=m

var x( ) = 1 n σ 2

varx

()

=

1

n

s

2

f x( ) = cx2, if x ≤1 0, otherwise

⎧ ⎨ ⎪

⎩⎪

fx

()

=

cx

2

, if x£1

0, otherwise

ì

í

ï

î

ï

E X( )

EX

()

var X( )

varX

()

θ

q

±0.03

±0.03

f x, y( ) = 6xy, 0 ≤ x ≤1; 0 ≤ y ≤ x

0, otherwise

⎧ ⎨ ⎪

⎩⎪

fx,y

()

=

6xy, 0£x£1; 0£y£x

0, otherwise

ì

í

ï

î

ï

E X Y( )

EXY

()

var X Y( )

varXY

()

X1( )

X

1

()

X2( )

X

2

()

di = X1i − X2i

d

i

=X

1i

-X

2i

µ

m

σ 2

s

2

E d( )

Ed

()

var d( )

vard

()

H0 :µ ≥10,000

H

0

:m³10,000

H0 :µ <10,000

H

0

:m<10,000

W = Xi i=1

5

∑

W=X

i

i=1

5

å

Xi ~ iid N µ,σ 2( )

X

i

~iidNm,s

2

()

E W( )

EW

()

var W( )

varW

()

µ̂ = X1 + X2 + X3 + X4 + X5

ˆ

m=X

1

+X

2

+X

3

+X

4

+X

5

µ

m

χ2

c

2

2x3 dx 0

1

∫

2x

3

dx

0

1

ò

α = 0.05

a=0.05

dy dx

dy

dx