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MATH1005 Quizzes

Quiz 7: Sampling distributions
Question 1 Questions
Suppose X1,,Xn are independent and each Xi has mean μ and variance σ2. If X¯ = 1 n i=1nX i, what is the distribution of X¯ when n is large ? Exactly one option must be correct)
a)
N nμ, σ2 n
b)
N nμ,σ2
c)
N μ, σ2 n
d)
N μ,σ2

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
Note that for large n we use the central limit theorem.
Choice (d) is incorrect
Let X1,,Xn be a random sample from a population with mean μ and variance σ2. If X¯ = 1 n i=1nX i is the sample mean, then what is the standard deviation of X¯ ? Exactly one option must be correct)
a)
σ n
b)
σ2 n
c)
σ n
d)
σ

Choice (a) is correct!
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
Suppose X1,,Xn are independent and Xi N(5,32) for i = 1,,n. If X¯ = 1 n i=1nX i, what is the distribution of X¯ ? Exactly one option must be correct)
a)
X¯ N(5,32)
b)
X¯ N 5, 32 n
c)
X¯ N 5, 3 n2
d)
X¯ N 5, 3 n

Choice (a) is incorrect
Choice (b) is correct!
Each Xi has mean 5 and variance 32 hence X¯ has mean 5 and variance 32 n and so X¯ N 5, 32 n .
Choice (c) is incorrect
Choice (d) is incorrect
Suppose X1,,Xn are independent and Xi N(3,52) for i = 1,,n. If X¯ = 1 n i=1nX i, what is the distribution of X¯ when n = 25 ? Exactly one option must be correct)
a)
X¯ = N(3,1)
b)
X¯ = N(3,5)
c)
X¯ = N(3,25)
d)
X¯ = N(5,1)

Choice (a) is correct!
Each Xi has mean 3 and variance 52 so X¯ N 3, 52 25 = N(3,1).
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
Consider the random variable X with distribution N(3,1). What is the value of P(X > 3.6)? Exactly one option must be correct)
a)
0.4522
b)
0.6058
c)
0.7258
d)
0.2742

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!
P(X¯ > 3.6) = P X μ σ > 3.6 μ σ = P(Z > 0.6) = 1 P(Z < 0.6) = 1 0.7258. = 0.2742.
Suppose X1,,Xn are independent and Xi N(5,32) for i = 1,,n. If X¯ = 1 n i=1nX i, what is the value of P(X¯ > 2) if n = 36 ? Exactly one option must be correct)
a)
0.99995
b)
< 0.0001
c)
0.8413
d)
0.1587

Choice (a) is correct!
X¯ N(5,0.25) = N(5,(0.5)2).
P(X¯ > 2) = P X¯ μ σ > 2 μ σ = P(Z > 6) = P(Z < 6) 0.99995,using tables.
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
A fair coin is tossed 30 times. The best approximation for the distribution of the number of heads is Exactly one option must be correct)
a)
N(30, 1 2)
b)
N(15,7.5)
c)
N(30,7.5)
d)
N 15, 7.5 30

Choice (a) is incorrect
Choice (b) is correct!
X B(30,0.5) N(np,np(1 p)) = N(15,7.5).
Choice (c) is incorrect
Choice (d) is incorrect
Suppose X¯ N 15, 7.5 30 . What is P(X¯ > 18) ? Exactly one option must be correct)
a)
Φ(3)
b)
1 Φ(3)
c)
Φ(3)
d)
1 Φ(6)

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!
X¯ N(15,.52).
Hence, P(X¯ > 18) = P Z > 18 15 0.5
= P(Z > 6) = 1 P(Z < 6) = 1 Φ(6).
Note that there are no continuity corrections for X¯.
Suppose X B(30,0.5). Use a normal approximation for B to find P(X > 20). Exactly one option must be correct)
a)
Φ(2.008)
b)
1 Φ(1.643)
c)
1 Φ(2.008)
d)
1 Φ(1.826)

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
X B(30,0.5), Y N(15,7.5) P(X > 20) = P(X 21)(use forbinomialdistribution) = P(Y > 20.5)(continuitycorrection) = P Z > 20.5 15 7.5 = P(Z > 2.008) = 1 P(Z < 2.008) = 1 Φ(2.008).
Choice (d) is incorrect
Suppose X B(30,0.5). Use a normal approximation for B to find P(X < 13). Exactly one option must be correct)
a)
Φ(0.913)
b)
Φ(0.548)
c)
1 Φ(0.913)
d)
1 Φ(0.548)

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
X B(30,0.5), Y N(15,7.5) P(X < 13) = P(X 12)(wemustuse forabinomialdistribution) = P(Y < 12.5)(continuitycorrection) = P Z < 12.5 15 7.5 = P(Z < 0.913) = P(Z > 0.913) = 1 P(Z < 0.913) = 1 Φ(0.913).
Choice (d) is incorrect