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Quiz 7: Sampling distributions

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Question 1

Suppose X1,…,Xn are independent and each Xi has mean μ and variance σ2. If -- 1∑n X = n Xi i=1 , what is the distribution of X when n is large ?

a)
( σ2) N nμ, n--
  b)
N (nμ,σ2)
c)
( σ2) N μ,n--
  d)
( 2) N μ,σ

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Your answer is correct.
Note that for large n we use the central limit theorem.
Not correct. Choice (d) is false.

Question 2

Let X1,…,Xn be a random sample from a population with mean μ and variance σ2. If -- 1∑n X = n- Xi i=1 is the sample mean, then what is the standard deviation of X ?

a)
√σ-- n
  b)
2 σ-- n
c)
σ n-
  d)
σ

 

Your answer is correct.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.

Question 3

Suppose X1,…,Xn are independent and Xi ~ N(5,32) for i = 1,…,n. If -- 1∑n X = n- Xi i=1 , what is the distribution of X ?

a)
X-~ N (5,32)
  b)
-- ( 32) X ~ N 5,n-
c)
( ) -- ( 3)2 X ~ N 5, n-
  d)
-- ( 3 ) X ~ N 5,√n--

 

Not correct. Choice (a) is false.
Your answer is correct.
Each Xi has mean 5 and variance 32 hence X has mean 5 and variance 32 n and so ( ) -- 32 X ~ N 5,n .
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.

Question 4

Suppose X1,…,Xn are independent and Xi ~ N(3,52) for i = 1,…,n. If -- ∑n X = 1- Xi n i=1 , what is the distribution of X when n = 25 ?

a)
-- X = N (3,1)
  b)
-- X = N(3,5)
c)
X- = N(3,25)
  d)
X- = N(5,1)

 

Your answer is correct.
Each Xi has mean 3 and variance 52 so X ~ N( 2) 3, 5- 25 = N(3,1).
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.

Question 5

Consider the random variable X with distribution N(3,1). What is the value of P(X > 3.6)?

a)
0.4522
   b)
0.6058
c)
0.7258
   d)
0.2742

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Your answer is correct.
-- ( ) P(X > 3.6) = P X---μ-> 3.6--μ = P (Z > 0.6) σ σ = 1- P(Z < 0.6) = 1- 0.7258. = 0.2742.

Question 6

Suppose X1,…,Xn are independent and Xi ~ N(5,32) for i = 1,…,n. If n X- = 1∑ X n i=1 i , what is the value of P(X > 2) if n = 36 ?

a)
≈ 0.99995
   b)
< 0.0001
c)
0.8413
   d)
0.1587

 

Your answer is correct.
-- 2 X ~ N(5,0.25) = N (5,(0.5) ).
( -- ) -- X---μ- 2--μ- P(X > 2) = P σ > σ = P (Z > - 6) = P (Z < 6) ≥ 0.99995,using tables.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.

Question 7

A fair coin is tossed 30 times. The best approximation for the distribution of the number of heads is

a)
N (30, 1) 2
  b)
N(15,7.5)
c)
N(30,7.5)
   d)
( ) N 15, 7.5 30

 

Not correct. Choice (a) is false.
Your answer is correct.
X ~ B(30,0.5) ~ N (np,np(1- p)) = N (15,7.5).
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.

Question 8

Suppose -- ( 7.5) X ~ N 15,30 . What is P(X > 18) ?

a)
Φ(3)
   b)
1 - Φ(3)
c)
Φ(3)
   d)
1 - Φ(6)

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Your answer is correct.
X ~ N(15,.52).
Hence, P(X > 18) = P( 18---15) Z > 0.5
= P(Z > 6) = 1 - P(Z < 6) = 1 - Φ(6).
Note that there are no continuity corrections for X.

Question 9

Suppose X ~ B(30,0.5). Use a normal approximation for B to find P(X > 20).

a)
Φ(2.008)
   b)
1 - Φ(1.643)
c)
1 - Φ(2.008)
   d)
1 - Φ(1.826)

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Your answer is correct.
X ~ B(30,0.5), Y ~ N(15,7.5)
P(X > 20) = P(X ≥ 21) (use ≥ for binomial distribution) = P(Y > 20.5) (continuity correction) ( 20.5--15) = P Z > √7.5- = P(Z > 2.008) = 1- P (Z < 2.008) = 1- Φ (2.008).
Not correct. Choice (d) is false.

Question 10

Suppose X ~ B(30,0.5). Use a normal approximation for B to find P(X < 13).

a)
Φ(0.913)
   b)
Φ(0.548)
c)
1 - Φ(0.913)
   d)
1 - Φ(0.548)

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Your answer is correct.
X ~ B(30,0.5), Y ~ N(15,7.5)
P(X < 13) = P (X ≤ 12) (we must use ≤ for a binomial distribution) = P (Y( < 12.5) (cont)inuity correction) = P Z < 12√.5--15- = P (Z < - 0.913) 7.5 = P (Z > 0.913) = 1 - P (Z < 0.913) = 1 - Φ (0.913).
Not correct. Choice (d) is false.
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