Week 10 - practice problems solutions Exercise 1nchristo/statistics403/stat403_week10...Title stat403_week10_practice_sol_w21x.pdf Created Date 3/12/2021 12:11:03 AM

1.13

Week 10 - practice problems solutions

Exercise 1

Exercise 2

b. A confidence interval is unbiased if the expected value of the interval midpoint is equal to the estimated parameter. For example

the midpoint of the interval x̄±z↵2

�pn

is x̄, and E(x̄) = µ. Now consider the confidence interval for �2. Show that the expected

value of the midpoint of this confidence interval is not equal to �2.

Answer:The midpoint of the confidence interval for �

2is:

1

2

(n � 1)s

2

�21�↵

2;n�1

+(n � 1)s

2

�2↵2

;n�1

�

and its expected value is:

1

2E

(n � 1)s

2

�21�↵

2;n�1

+(n � 1)s

2

�2↵2

;n�1

�=

1

2

(n � 1)E(s

2)

�21�↵

2;n�1

+(n � 1)E(s

2)

�2↵2

;n�1

�.

Since E(s2) = �

2the expression above is:

1

2�2

(n � 1)

�21�↵

2;n�1

+(n � 1)

�2↵2

;n�1

�6= �

2

because

(n � 1)

�21�↵

2;n�1

+(n � 1)

�2↵2

;n�1

�6= 2.

Exercise 4Let X be a uniform random variable on (0, ✓). You have exactly one observation from this distribution and you want to test the null

hypothesis H0 : ✓ = 10 against the alternative Ha : ✓ > 10, and you want to use significance level ↵ = 0.10. Two testing procedures

are being considered:

Procedure G rejects H0 if and only if X � 9.

Procedure K rejects H0 if either X � 9.5 or if X 0.5.

a. Confirm that Procedure G has a Type I error probability of 0.10.

Answer:

↵ = P (X � 9|✓ = 10)

Z 10

9

1

10dx =

x

10|109 = 0.10.

b. Confirm that Procedure K has a Type I error probability of 0.10.

Answer:

↵ = P (X � 9.5|✓ = 10) + P (X 0.5|✓ = 10) =

Z 10

9.5

1

10dx +

Z 0.5

0

1

10dx =

x

10|109.5 +

x

10|0.50 = 0.10.

c. Find the power of Procedure G when ✓ = 12.

Answer:

1 � � = P (X � 9|✓ = 12)

Z 12

9

1

12dx =

x

12|129 = 0.25.

d. Find the power of Procedure K when ✓ = 12.

Answer:

1 � � = P (X � 9.5|✓ = 12) + P (X 0.5|✓ = 12) =

Z 12

9.5

1

12dx +

Z 0.5

0

1

12dx =

x

12|129.5 +

x

12|0.50 = 0.25.

Exercise 3Exercise 5Exercise 3

Exercise 8Exercise 4