Question

Give asymptotic upper and lower bounds for T .n/ in each of the following recurrences. Assume that T .n/ is constant for sufficiently small n. Make your bounds as tight as possible, and justify your answers.

a) T(n) = T(n/2) +T(n/4)+T(n/8)+n

b) T(n) = T(n-1) +1/n

c) T(n)= T(n-1) +lg n

d) T(n) = T(n-2) +1/lgn

Answer #1

**a.** T(n) = T(n/2) +T(n/4)+T(n/8)+n

**Answer:**

**b.** T(n) = T(n-1) +1/n

**Answer:**

**c.** T(n)= T(n-1) +lg n

**Answer:**

**d.** T(n) = T(n-2) +1/lgn

**Answer:**

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Study guide practice problem:
1)Find and prove the tight asymptotic bounds (theta) for the
following recurrence equations. You can make an appropriate
assumption for T(1), or T(0):
a) T(n)=T(n-1)+3
b) T(n)=T(n/2)+1

Use a recursive tree method to compute a tight asymptotic upper
bound for recurrence function T(n)= 3T(n/4)+n
.
then use substitution method to verify your answer.

Use Master Theorem to solve the following recurrences. Justify
your answers.
(1) T(n) = 3T(n/3) + n
(2) T(n) = 8T(n/2) + n^2
(3) T(n) = 27T(n/3) + n^5
(4) T(n) = 25T(n/5) + 5n^2

Determine the p-value for each of the following
situations. (Give your answer bounds exactly.)
(a) Ha: β1 > 0, with
n = 25 and t = 1.2
< p <
(b) Ha: β1 ≠ 0, with
n = 22, b1 = 0.22, and
sb1 = 0.08
< p <
(c) Ha: β1 < 0, with
n = 22, b1 = -1.46, and
sb1 = 0.79
< p <

Determine the p-value for each of the following situations.
(Give your answer bounds exactly.) (a) Ha: β1 > 0, with n = 29
and t = 1.76 < p < (b) Ha: β1 ≠ 0, with n = 15, b1 = 0.23,
and sb1 = 0.11 < p < (c) Ha: β1 < 0, with n = 25, b1 =
-1.53, and sb1 = 0.71

Analysing Algorithmic Efficiency (Marks: 3)
Analyze the following code fragment and provide an asymptotic
(Θ) bound on the running time as a function of n. You do not need
to give a formal proof, but you should justify your answer.
1: foo ← 0
2: for i ← 0 to 2n 2 do
3: foo ← foo × 4
4: for j ← 1396 to 2020 do
5: for k ← 4i to 6i do
6: foo ← foo ×...

For each set of conditions below, give an example of a predicate
P(n) deﬁned on N that satisfy those conditions (and justify your
example), or explain why such a predicate cannot exist.
(a) P(n) is True for n ≤ 5 and n = 8; False for all other
natural numbers.
(b) P(1) is False, and (∀k ≥ 1)(P(k) ⇒ P(k + 1)) is True.
(c) P(1) and P(2) are True, but [(∀k ≥ 3)(P(k) ⇒ P(k + 1))] is
False....

Determine the upper-tail critical value t Subscript alpha
divided by 2tα/2 in each of the following circumstances. a.a. 1
minus alpha equals 0.99 comma n equals 81−α=0.99, n=8 d.d. 1 minus
alpha equals 0.99 comma n equals 591−α=0.99, n=59 b.b. 1 minus
alpha equals 0.95 comma n equals 81−α=0.95, n=8 e.e. 1 minus alpha
equals 0.90 comma n equals 531−α=0.90, n=53 c.c. 1 minus alpha
equals 0.99 comma n equals 441−α=0.99, n=44?

Is there a set A ⊆ R with the following property? In each case
give an example, or a rigorous proof that it does not exist.
d) Every real number is both a lower and an upper bound for
A.
(e) A is non-empty and 2inf(A) < a < 1 sup(A) for every a
∈ A.2
(f) A is non-empty and (inf(A),sup(A)) ⊆ [a+ 1,b− 1] for some
a,b ∈ A and n > 1000.

Assume that you have a sample of n 1 equals 5, with the sample
mean Upper X overbar 1 equals 48, and a sample standard deviation
of Upper S 1 equals 6, and you have an independent sample of n 2
equals 4 from another population with a sample mean of Upper X
overbar 2 equals 30 and the sample standard deviation Upper S 2
equals 7. Assuming the population variances are equal, at the 0.01
level of significance, is...

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