The Basic Practice of Statistics
The Basic Practice of Statistics
8th Edition
ISBN: 9781319057916
Author: Moore
Publisher: MAC HIGHER
Question
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Chapter 18, Problem 18.52E

(a)

To determine

To find: The probability of a Type I error.

(a)

Expert Solution
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Answer to Problem 18.52E

The probability of a Type I error is 0.5.

Explanation of Solution

Calculation:

The probability of Type I error:

Formula for probability of Type I error is,

P(Type I error)=P(x¯>0)|μ=0=P(x¯μ(σn)>0μ(σn))|μ=0=P(Z>00(σn))

                                 =P(Z>0)=1P(Z0)

From Table A: Standard Normal Cumulative Proportions, the value of P(Z0) is 0.5.

Therefore,

P(Type I error)=1P(Z0)=10.5=0.5

Thus, the probability that Type I error is 0.5.

(b)

To determine

To find: The probability of a Type II error when μ=0.5 .

(b)

Expert Solution
Check Mark

Answer to Problem 18.52E

The probability of a Type II error when μ=0.5 is 0.1587.

Explanation of Solution

Calculation:

Probability of Type II error:

P(Type II error)=P(x¯0)|μ=0.5=P(x¯μ(σn)0μ(σn))|μ=0.5=P(Z00.5(2.525))=P(Z1)

From Table A: Standard Normal Cumulative Proportions, the value of P(Z1) is 0.1587.

Therefore,

P(Type II error)=0.1587

Thus, the probability of a Type II error when μ=0.5 is 0.1587.

(c)

To determine

To find: The probability of a Type II error when μ=1.0 .

(c)

Expert Solution
Check Mark

Answer to Problem 18.52E

The probability of a Type II error when μ=1.0 is 0.0228.

Explanation of Solution

Calculation:

Probability of Type II error:

P(Type II error)=P(x¯0)|μ=1.0=P(x¯μ(σn)0μ(σn))|μ=1.0=P(Z01.0(2.525))=P(Z2)

From Table A: Standard Normal Cumulative Proportions, the value of P(Z2) is 0.0228.

Therefore,

P(Type II error)=0.0228

Thus, the probability of a Type II error when μ=1.0 is 0.0228.

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This problem is based on the fundamental option pricing formula for the continuous-time model developed in class, namely the value at time 0 of an option with maturity T and payoff F is given by: We consider the two options below: Fo= -rT = e Eq[F]. 1 A. An option with which you must buy a share of stock at expiration T = 1 for strike price K = So. B. An option with which you must buy a share of stock at expiration T = 1 for strike price K given by T K = T St dt. (Note that both options can have negative payoffs.) We use the continuous-time Black- Scholes model to price these options. Assume that the interest rate on the money market is r. (a) Using the fundamental option pricing formula, find the price of option A. (Hint: use the martingale properties developed in the lectures for the stock price process in order to calculate the expectations.) (b) Using the fundamental option pricing formula, find the price of option B. (c) Assuming the interest rate is very small (r ~0), use Taylor…
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