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You have been asked to compare two neighbourhoods in the west end of Montreal by a real estate agency. The agency wants to know if a difference exists in the selling price of houses on Westminster street (Montreal West), as compared to Beaconsfield street (NDG). You take a sample of houses sold on both streets over the past 10 years, adjust the inflation and monetary rate so as to equalize the dollar, and tabulate your results. Is there enough evidence to prove that the selling price for the houses on the two streets is significantly different? Please conduct a complete hypothesis test at the 99% confidence level and calculate the p-value. (Hint: Can we assume that the variances are equal?)
Street n x s
Westminister 16 125,950 2400
Beaconsfield 24 128,800 3700
Street n x s
Westminister 16 125,950 2400
Beaconsfield 24 128,800 3700
A group of ten students from the course INTE 296 was approached by a famous scientist by the name of Dr. Wild E. Coyote with an experimental drug to aid in their retention of statistical procedures. The drug, called MemoraidTM, was administered after the students completed a generic, comprehensive exam. Twenty-four hours later, the students were given a similar exam, and the test scores were compared. Based on the following table, is there enough evidence to conclude that the drug was effective in increasing their statistical ability? Use an alpha value of 0.05 and find the p-value.
Student: 1 2 3 4 5 6 7 8 9 10
Before: 75 62 70 70 55 59 60 64 72 65
After: 74 65 68 74 62 62 60 67 75 58
Student: 1 2 3 4 5 6 7 8 9 10
Before: 75 62 70 70 55 59 60 64 72 65
After: 74 65 68 74 62 62 60 67 75 58
Please find the limit(s) of the boundaries for the critical region for the following hypotheses (You only need to find the z, t, or F-critical value in all cases, e.g., +/2.33. Please do not complete the hypothesis test):
a. Ho: μd ≤ 6.5, Ha: μd > 6.5, α = 0.05, n = 35, s = 2.5.
b. Ho: μ1 - μ2 ≤ 0, Ha: μ1 - μ2 > 0, α = 0.1, n1 = 13, n2 = 17 (σ1 = σ2).
c. Ho: μ1 - μ2 ≥ 0, Ha: μ1 - μ2 < 0, α = 0.025, n1 = 19, n2 = 27 (σ1 ≠ σ2).
d. Ho: σ12 = σ22, Ha: σ12 ≠ σ22, α = 0.01, n1 = 31, n2 = 21 s1 = 4.1, s2 = 8.3.
e. Ho: σ12 = σ22, Ha: σ12 ≠ σ22, α = 0.05, n1 = 22, n2 = 13 s1 = 7.2, s2 = 10.6.
a. Ho: μd ≤ 6.5, Ha: μd > 6.5, α = 0.05, n = 35, s = 2.5.
b. Ho: μ1 - μ2 ≤ 0, Ha: μ1 - μ2 > 0, α = 0.1, n1 = 13, n2 = 17 (σ1 = σ2).
c. Ho: μ1 - μ2 ≥ 0, Ha: μ1 - μ2 < 0, α = 0.025, n1 = 19, n2 = 27 (σ1 ≠ σ2).
d. Ho: σ12 = σ22, Ha: σ12 ≠ σ22, α = 0.01, n1 = 31, n2 = 21 s1 = 4.1, s2 = 8.3.
e. Ho: σ12 = σ22, Ha: σ12 ≠ σ22, α = 0.05, n1 = 22, n2 = 13 s1 = 7.2, s2 = 10.6.
c) Given a random sample n X , X , , X 1 2
K from a normal ( , ) 2 N μ s distribution,
examine unbiasedness and consistency of (i) X for μ , (ii) − 2 ( )
1
X X
n i
for 2 s .
K from a normal ( , ) 2 N μ s distribution,
examine unbiasedness and consistency of (i) X for μ , (ii) − 2 ( )
1
X X
n i
for 2 s .
b) For the geometric distribution,
( , ) (1 ) , 1, 2, , 0 1 1 q = q − q = < q < f x − x K x
Obtain an unbiased estimator of 1/ q
( , ) (1 ) , 1, 2, , 0 1 1 q = q − q = < q < f x − x K x
Obtain an unbiased estimator of 1/ q
9. a) If x ³1 is the critical region for testing : 2 0 H q = against the alternative q = 1, on
the basis of the single observation from the population,
f (x, q) = qexp(−q, x), 0 £ x < ¥
Obtain the values of type I and type II errors.
the basis of the single observation from the population,
f (x, q) = qexp(−q, x), 0 £ x < ¥
Obtain the values of type I and type II errors.
Math
b) Solve the following game using dominance principle. (5)
Player B
B1 B2 B3 B4
A1
3 2 4 0
Player A A2
3 4 2 4
A3
4 2 4 0
A4
0 4 0 8
6. For the following payoff matrix, transform the zero-sum game into an equivalent linear
programming problem and solve it by using simplex method. (10)
Player B
B1 B2 B3
A1
1 −1 3
Player A A2
3 5 − 3
A3
6 2 − 2 6
Player B
B1 B2 B3 B4
A1
3 2 4 0
Player A A2
3 4 2 4
A3
4 2 4 0
A4
0 4 0 8
6. For the following payoff matrix, transform the zero-sum game into an equivalent linear
programming problem and solve it by using simplex method. (10)
Player B
B1 B2 B3
A1
1 −1 3
Player A A2
3 5 − 3
A3
6 2 − 2 6
Math
7. a) Solve the following LPP using two phase method:
1 2 3 max Z = 2x + 3x +10x
Subject to:
0 2 x1 + x3 =
1 x2 + x3 =
0 , , x1
x2
x3 ≥ (6)
b) Which of the following sets are convex? Give reason.
i) }0 {( , :) ;1 , A = x1
x2
x1
x2 ≤ x1
x2 ≥
ii) }0 {( , :) 3 ; ,
1 2
2 B = x1
x2
x2 − ≥ x1
1 2 3 max Z = 2x + 3x +10x
Subject to:
0 2 x1 + x3 =
1 x2 + x3 =
0 , , x1
x2
x3 ≥ (6)
b) Which of the following sets are convex? Give reason.
i) }0 {( , :) ;1 , A = x1
x2
x1
x2 ≤ x1
x2 ≥
ii) }0 {( , :) 3 ; ,
1 2
2 B = x1
x2
x2 − ≥ x1
Math
8. a) Without sketching the region, check whether ) P ,0( 0 is in the convex hull of the
points )0 A(− ,1 − ),1 B ,1( and )1,0( C . If it is in the region, write P as convex
combination of A, and B C . (4)
b) Use simplex method to solve the following LP problem:
Maximize 1 2 3 Z = 3x + 5x + 4x
Subject to:
8 2 3 x1 + x2 ≤
10 2 5 x2 + x3 ≤
15 3 2 4 x1 + x2 + x3 ≤
0
points )0 A(− ,1 − ),1 B ,1( and )1,0( C . If it is in the region, write P as convex
combination of A, and B C . (4)
b) Use simplex method to solve the following LP problem:
Maximize 1 2 3 Z = 3x + 5x + 4x
Subject to:
8 2 3 x1 + x2 ≤
10 2 5 x2 + x3 ≤
15 3 2 4 x1 + x2 + x3 ≤
0
Math
9. a) Let 4 ,2 x1 = x2 = and 1 x3 = be a feasible solution to the system of equations
2 2 2 x1 − x2 + x3 =
18 4 x1 + x2 =
Reduce the given feasible solution to a basic feasible solution. (5)
b) Compute all the basic feasible solutions to the L.P. problem
Maximize 1 2 3 Z = 3x + 2x + x
Subject to
8 3 2 2 x1 + x2 + x3 + x4 =
7 3 4 x1 + x2 + x3 + x5 =
and
0 , , , , x1
x2
x3
x4
x5 ≥
2 2 2 x1 − x2 + x3 =
18 4 x1 + x2 =
Reduce the given feasible solution to a basic feasible solution. (5)
b) Compute all the basic feasible solutions to the L.P. problem
Maximize 1 2 3 Z = 3x + 2x + x
Subject to
8 3 2 2 x1 + x2 + x3 + x4 =
7 3 4 x1 + x2 + x3 + x5 =
and
0 , , , , x1
x2
x3
x4
x5 ≥
"assignmentexpert.com" is professional group of people in Math subjects! They did assignments in very high level of mathematical modelling in the best quality. Thanks a lot
#339914 on Dec 2023
#339914 on Dec 2023