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Quiz 6: Normal distributions

Last unanswered question  Question  Next unanswered question
 

Question 1

 
 
Which of the following random variables would you expect to be discrete?
a) The weights of mechanically produced items.
b) The number of children at a christmas party.
c) The lifetimes of resistors.
d) The distance between Centre Point tower and any point on the Australian continent.
e) The times, in seconds, for a 100m sprint.

 

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

Question 2

 
 
Let X ~ N(3,22). What does this tell us about the distribution of X ?
a) X is binomial with n = 3 and p = 2.
b) X is normal with mean 3 and variance 4.
c) X is normal with mean 3 and variance 2.
d) X is binomial with mean 2 and variance 9.

 

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

Question 3

 
 
X is a random normal variable, with mean μ and variance σ2. The “standardised form” of X is Z = X--μ σ.
What are the mean and variance, respectively, of Z ?
a) 0,1   b) 2,1
c) 1,0   d) 2,0

 

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

Question 4

 
 
Let X ~ N(5,32). Which of the following is a standard normal variable ?
a) Z = X---5- 5   b) X - 3 Z = ------ 5
c) X----5 Z = 3   d) Z = X----3 3

 

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

Question 5

 
 
Suppose X ~ N(5,32). What is P(X ≤ 8) in terms of the standard normal variable Z ?
a) P(Z ≤ 1)   b) P(Z ≤-1)
c) P(Z ≤ 0.6)   d) P(Z ≤-0.6)
e) P(Z ≤ 1.67)

 

Your answer is correct.
(X - μ 8- μ) P (X ≤ 8) = P --σ---≤ -σ--- = P (Z ≤ 33) = P(Z ≤ 1) .
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.
Not correct. Choice (e) is false.
 

Question 6

 
 
Suppose X ~ N(5,32). What is the value of P(X ≤ 2) ?
a) 0.8413   b) 0.1587
c) 0.7258   d) 0.2742

 

Not correct. Choice (a) is false.
Your answer is correct.
P(X ≤ 2) = P( ) 2---5 Z ≤ 3 = P(Z ≤-1) = 1 - P(Z ≤ 1)
= 1 - 0.8413 = 0.1587.
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.
 

Question 7

 
 
Suppose X is normally distributed with mean 5 and standard deviation 0.4 . Using the standard transformation Z = X---μ- σ we find P(X ≤ X0) = P(Z ≤ 1.3). What is the value of X0 ?
a) 6.9   b) 4.48
c) 2.00   d) 5.52

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Your answer is correct.
X0---5 0.4 = 1.3, X0 - 5 = 0.52, X = 5.52. 0
 

Question 8

 
 
Suppose X is normally distributed with mean 5. If P(X ≤ 3) = 0.2 what is the standard deviation of X ?
a) σ = 0.42   b) σ = -0.42
c) σ = 0.38   d) σ = 2.38

 

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) = P( ) Z ≤ 3---5 σ = 0.2
therefore,P( ) 5--3- Z ≤ σ = 0.8.
But P(Z ≤ 0.84) = 0.8 ⇒5--3- σ = 0.84 ⇒ σ = 2.38.
 

Question 9

 
 
Suppose that X ~ N(2,1) and Y ~ N(3,2). Assuming X and Y are independent what is the distribution of X + Y ?
a) N(3,5)   b) N(5,3)
c) N(3,3)   d) N(5,5)

 

Not correct. Choice (a) is false.
Your answer is correct.
The mean of X + Y is 2 + 3 = 5. The variance of X + Y is 1 + 2 = 3.
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.
 

Question 10

 
 
X and Y are independent random variables. The mean and variance of X are 2 and 1 respectively. The mean and variance of Y are 3 and 2 respectively.
 
Which of the statements below about the random variable X - Y is true ?
a) X - Y ~ N(-1,1)
b) X - Y has mean 5 and variance 3.
c) X - Y ~ N(-1,3)
d) X - Y has mean -1 and variance 3.

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Your answer is correct.
We do not know that X and Y are normally distributed. The mean of X - Y is 2 - 3 = -1. The variance of X - Y is 1 + (-1)2 2 = 3.