Let X space tilde thin space U left parenthesis 0 comma 20 right parenthesis and assume we take a sample of size 100 from this population distribution. What distribution would the sample mean (top enclose X space equals space 1 over 100 space sum from i equals 1 to 100 of space X subscript i ) follow?

Selected Answer:  
b.
top enclose X space tilde thin space N left parenthesis 10 comma 0.333 right parenthesis approximately according to the central limit theorem.

Answers:  
a.
top enclose X space tilde U left parenthesis 0 comma 2 right parenthesis

b.
top enclose X space tilde thin space N left parenthesis 10 comma 0.333 right parenthesis approximately according to the central limit theorem.

c.
top enclose X space tilde thin space N left parenthesis 10 comma 3.33 right parenthesis approximately according to the central limit theorem.

d.
top enclose X space tilde space U left parenthesis 0 comma 20 right parenthesis

Question 2
Let X = the amount of cereal packed into a box and assume we know X space tilde thin space N left parenthesis 500 comma space 8 squared right parenthesis . If we take a sample of size 10, what distribution will the sample mean (top enclose X space equals space 1 over 10 space sum from i equals 1 to 10 of X subscript i ) follow?

Selected Answer:  
c.
N left parenthesis 500 comma 64 right parenthesis

Answers:  
a.
N left parenthesis 500 comma 6.4 right parenthesis

b.
N left parenthesis 500 comma bevelled fraction numerator 64 over denominator square root of 10 end fraction right parenthesis

c.
N left parenthesis 500 comma 64 right parenthesis

d.
N left parenthesis 500 comma 640 right parenthesis

Question 3
Let X subscript 1 comma space X subscript 2 comma space X subscript 3 comma space horizontal ellipsis comma space X subscript n space be iid random variables with X subscript i space tilde thin space N left parenthesis mu comma sigma squared right parenthesis , and define the usual sample mean ie top enclose X space equals space 1 over n space sum from i equals 1 to n of X subscript i

We know that if Z space equals space fraction numerator top enclose X space minus space mu over denominator begin display style bevelled fraction numerator sigma over denominator square root of n end fraction end style end fraction then Z space tilde thin space N left parenthesis 0 comma 1 right parenthesis .

If we now define W space equals space fraction numerator top enclose X space minus space mu over denominator begin display style bevelled fraction numerator S over denominator square root of n end fraction end style end fraction where S comes from the usual estimator of sigma , ie S squared space equals space fraction numerator 1 over denominator n minus 1 end fraction space space sum from i space equals space 1 to n of left parenthesis X subscript i space minus space top enclose X right parenthesis squared , then what distribution does W follow?

Selected Answer:  
a.
t subscript n minus 1 end subscript (ie a t-distribution with n-1 degrees of freedom)

Answers:  
a.
t subscript n minus 1 end subscript (ie a t-distribution with n-1 degrees of freedom)

b.
t subscript n (ie a t-distribution with n degrees of freedom)

c.
N(0,1.5)

d.
N(0,1)

Question 4
What is the relationship between Z (ie a standard normal distribution) and t subscript n (ie a t-distribution with n degrees of freedom)?

Selected Answer:  
Z and t subscript n are similar shape but t subscript n has skinnier tails (ie extreme values are less likely under t subscript n )

As n gets smaller t subscript n looks more and more like Z .

Answers:  
Z and t subscript n are similar shape but t subscript n has fatter tails (ie extreme values are more likely under t subscript n )

As n gets larger t subscript n looks more and more like Z .

Z and t subscript n are similar shape but t subscript n has skinnier tails (ie extreme values are less likely under t subscript n )

As n gets larger t subscript n looks more and more like Z .

Z and t subscript n are similar shape but t subscript n has fatter tails (ie extreme values are more likely under t subscript n ) .

As n gets smaller t subscript n looks more and more like Z .

Z and t subscript n are similar shape but t subscript n has skinnier tails (ie extreme values are less likely under t subscript n )

As n gets smaller t subscript n looks more and more like Z .

Question 5
If we are interested in the mean expenditure of Australians on gambling, which of the following is the correct use of symbols and terminology as per the chapter on the distribution of estimators?

Selected Answer:  
a.
mu is an estimate of the population parameter top enclose x .

E left parenthesis mu right parenthesis space equals top enclose X or in other words mu is biased.

Answers:  
a.
mu is an estimate of the population parameter top enclose x .

E left parenthesis mu right parenthesis space equals top enclose X or in other words mu is biased.

b.
top enclose x is an estimate of the population parameter mu .

E left parenthesis top enclose X right parenthesis space equals space mu or in other words top enclose X is unbiased.

c.
mu is an estimate of the population parameter top enclose x .

E left parenthesis mu right parenthesis space equals top enclose X or in other words mu is unbiased.

d.
top enclose x is an estimate of the population parameter mu .

E left parenthesis top enclose X right parenthesis space equals space mu or in other words top enclose X is biased.