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The U.S. Coast Guard (USCG) provides a wide variety of information on boating accidents including the wind condition at the time of the accident. The following table shows the results obtained for 4401 accidents (USCG website, November 8, 2012). Wind Condition Percentage of Accidents None 9.6 Light 57.0 Moderate 23.8 Strong 7.7 Storm 1.9 Let x be a random variable reflecting the known wind condition at the time of each accident. Set x = 0 for none, x = 1 for light, x = 2 for moderate, x = 3 for strong, and x = 4 for storm. a. Develop a probability distribution for x . b. Compute the expected value of x . c. Compute the variance and standard deviation for x . Comment on what your results imply about the wind conditions during boating accidents.

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STATISTICS F/BUSINESS+ECONOMICS-TE...

13th Edition
Anderson
Publisher: CENGAGE L
ISBN: 9781305881884
BuyFind

STATISTICS F/BUSINESS+ECONOMICS-TE...

13th Edition
Anderson
Publisher: CENGAGE L
ISBN: 9781305881884

Solutions

Chapter
Section
Chapter 5, Problem 59SE
Textbook Problem

The U.S. Coast Guard (USCG) provides a wide variety of information on boating accidents including the wind condition at the time of the accident. The following table shows the results obtained for 4401 accidents (USCG website, November 8, 2012).

Wind Condition

Percentage of

Accidents

None 9.6
Light 57.0
Moderate 23.8
Strong 7.7
Storm 1.9

Let x be a random variable reflecting the known wind condition at the time of each accident. Set x = 0 for none, x = 1 for light, x = 2 for moderate, x = 3 for strong, and x = 4 for storm.

  1. a. Develop a probability distribution for x.
  2. b. Compute the expected value of x.
  3. c. Compute the variance and standard deviation for x.

Comment on what your results imply about the wind conditions during boating accidents.

Expert Solution

a.

To determine

Construct a probability distribution for the random variable x.

Answer to Problem 59SE

The probability distribution for the random variable x is given by,

xf(x)
00.0960
10.05700
20.2380
30.0770
40.0190

Explanation of Solution

Calculation:

The data represents the results obtained for 4,401 boating accidents including the wind condition at the time of the accident. The random variable x represents the known wind condition at the time of each accident. The random variable x takes the value 0 for none,

takes the value 1 for light, takes the value 2 for moderate, takes the value 3 for strong, takes the value 4 for storm.

Here, the total number of responses is 4,401. The corresponding probabilities are obtained by converting the percentages in to probabilities. That is, by dividing each value with 100.

The probability distribution for the random variable x can be obtained as follows:

xffNf(x)
09.69.61000.0960
157.057.01000.5700
223.823.81000.2380
37.77.71000.0770
41.91.91000.0190
Total100 1
Expert Solution

b.

To determine

Find the expected value for the random variable x.

Answer to Problem 59SE

The expected value for the random variable x is 1.353.

Explanation of Solution

Calculation:

The formula for the expected value of a discrete random variable is,

E(x)=μ=xf(x)

The expected value for the random variable x is obtained using the following table:

xf(x)xf(x)
00.0960
10.570.57
20.2380.476
30.0770.231
40.0190.076
Total11.353

Thus, the expected value for the random variable x is 1.353.

Expert Solution

c.

To determine

Find the variance and standard deviation of the random variable x.

Answer to Problem 59SE

The variance of the random variable x is 0.6884.

The standard deviation of the random variable x is 0.8297.

Explanation of Solution

Calculation:

The formula for the variance of the discrete random variable is,

σ2=[(xμ)2. f(x)]

The variance of the random variable x is obtained using the following table:

xf(x)(xμ)(xμ)2(x-μ)2. f(x)
00.096–1.3531.83060.1757
10.57–0.3530.12460.0710
20.2380.6470.41860.0996
30.0771.6472.71260.2089
40.0192.6477.00660.1331
Total13.23512.09300.6884

Therefore,

σ2=0.6884

Thus, the variance of the random variable x is 0.6884.

The formula for the standard deviation of the discrete random variable is,

σ=[(xμ)2. f(x)]

Thus, the standard deviation is,

σ=0.6884=0.8297

Hence, the standard deviation of the random variable x is 0.8297.

Expert Solution

d.

To determine

Explain what the result implies about the wind conditions during the boating accidents.

Explanation of Solution

The expected value is 1.353 and it represents the mean wind conditions when accident occurs. This value is slightly less than light wind conditions.

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Chapter 5 Solutions

STATISTICS F/BUSINESS+ECONOMICS-TEXT
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