Applied Statistics in Business and Economics
Applied Statistics in Business and Economics
5th Edition
ISBN: 9780077837303
Author: David Doane, Lori Seward Senior Instructor of Operations Management
Publisher: McGraw-Hill Education
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Chapter 5, Problem 96CE

High levels of cockpit noise in an aircraft can damage the hearing of pilots who are exposed to this hazard for many hours. Cockpit noise in a jet aircraft is mostly due to airflow at hundreds of miles per hour. This 3 × 3 contingency table shows 61 observations of data collected by an airline pilot using a handheld sound meter in a certain aircraft cockpit. Noise level is defined as “low” (under 88 decibels), “medium” (88 to 91 decibels), or “high” (92 decibels or more). There are three flight phases (climb, cruise, descent). (a) Calculate each probability (i–vi) and explain in words what it means. (b) Do you see evidence that noise level depends on flight phase? Explain. (c) Where else might ambient noise be an ergonomic issue? (Hint: Search the web.)

  1. i. P(B)
  2. ii. P(L)
  3. iii. P(H | C)
  4. iv. P(H | D)
  5. v. P(L and B)
  6. vi. P(L and C)

Chapter 5, Problem 96CE, High levels of cockpit noise in an aircraft can damage the hearing of pilots who are exposed to this

a.

Expert Solution
Check Mark
To determine

Calculate each probability (i-vi) and explain it meaning in words.

Answer to Problem 96CE

i. The probability P(B) is 0.4098 and the meaning is the probability of a climb is 0.4098.

ii. The probability P(L) is 0.2295 and the meaning is the probability of a low noise level is 0.2295.

iii. The probability P(H|C) is 0.3750 and the meaning is the probability of high noise given that the person is in the cruise phase is 0.3750.

iv. The probability P(H|D) is 0.5 and the meaning is the probability of high noise given that the person is in the descent phase is 0.5.

v. The probability P(L and B) is 0.0984 and the meaning is the probability of low noise and climbing is 0.0984.

vi. The probability P(L and C) is 0.0328 and the meaning is the probability of low noise and cruising is 0.0328.

Explanation of Solution

Calculation:

The given table shows that cockpit noise levels in the jet aircraft.

The given contingency table is,

 Flight Phase
Noise LevelClimb (B)Cruise (C)Descent (D)Row Total
Low (L)62614
Medium (M)183829
High (H)131418
Column Total2582861

For (i)P(B):

The formula for finding the probability P(B) is,

P(B)=Frequency for the class BTotal frequencies in the distribution

Substitute 25 for ‘Frequency for the class B’ and 61 for ‘Total frequencies in the distribution’

P(B)=2561=0.4098

Therefore, the probability P(B) is 0.4098 and the meaning is the probability of a climb is 0.4098.

For (ii) P(L):

The formula for finding the probability P(L) is,

P(L)=Frequency for the class LTotal frequencies in the distribution

Substitute 14 for ‘Frequency for the class L’ and 61 for ‘Total frequencies in the distribution’

P(L)=1461=0.2295

Therefore, the probability P(L) is 0.2295 and the meaning is the probability of a low noise level is 0.2295.

For (iii) P(H|C):

The formula for finding the probability P(H|C) is,

P(H|C)=Frequency for the class H and CFrequency for the class C

Substitute 3 for ‘Frequency for the class H and C’ and 8 for ‘Frequency for the class C’,

P(H|C)=38=0.3750

Therefore, the probability P(H|C) is 0.3750 and the meaning is the probability of high noise given that the person is in the cruise phase is 0.3750.

For (iv) P(H|D):

The formula for finding the probability P(H|D) is,

P(H|D)=Frequency for the class H and DFrequency for the class D

Substitute 14 for ‘Frequency for the class H and D’ and 28 for ‘Frequency for the class D’,

P(H|D)=1428=0.5

Therefore, the probability P(H|D) is 0.5 and the meaning is the probability of high noise given that the person is in the descent phase is 0.5.

For (v) P(L and B):

The formula for finding the probability P(L and B) is,

P(L and B)=Frequency for the class L and BTotal frequencies in the distribution

Substitute 6 for ‘Frequency for the class L and B’ and 61 for ‘Total frequencies in the distribution’,

P(L and B)=661=0.0984

Therefore, the probability P(L and B) is 0.0984 and the meaning is the probability of low noise and climbing is 0.0984.

For (vi) P(L and C):

The formula for finding the probability P(L and C) is,

P(L and C)=Frequency for the class L and CTotal frequencies in the distribution

Substitute 2 for ‘Frequency for the class L and C’ and 61 for ‘Total frequencies in the distribution’,

P(L and C)=261=0.0328

Therefore, the probability P(L and C) is 0.0328 and the meaning is the probability of low noise and cruising is 0.0328.

b.

Expert Solution
Check Mark
To determine

Check whether there is evidence that noise level depends on flight phase. Also, explain the reason.

Answer to Problem 96CE

Yes, there is evidence that noise level depends on flight phase because P(H and C)P(H)P(C).

Explanation of Solution

Calculation:

Special law of multiplication:

If two events A and B are independent, then

P(AB)=P(A)P(B)

Consider noise level as High and flight phase type as cruise.

The formula for checking independence is,

P(H and C)=P(H)P(C)(Frequency for the class H and CTotal frequencies in the distribution)=((Frequency for the class HTotal frequencies in the distribution)×(Frequency for the class C0Total frequencies in the distribution))361=1861×8610.0492=1443,721

                                                        0.04920.0387

Here, it is observed that P(H and C)P(H)P(C). Therefore, there is evidence that noise level depends on flight phase.

c.

Expert Solution
Check Mark
To determine

Identify the ambient noise is an ergonomic issue.

Explanation of Solution

Answers may vary: One of the answers is given below:

Some of the ambient noises are an ergonomic issues are,

  • • The noise from the heavy machines running in a factory.
  • • The excess sound created by vehicles in the traffic signals.

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