Question

The Transportation Security Administration (TSA) collects data on wait time at each of its airport security checkpoints. For flights departing from Terminal 3 at John F. Kennedy International Airport (JFK) between 3:00 and 4:00 PM on Wednesday, the mean wait time is 12 minutes, and the maximum wait time is 16 minutes. [Source: Transportation Security Administration, summary statistics based on historical data collected between February 18, 2008, and March 17, 2008.]

Assume that x, the wait time at the Terminal 3 checkpoint at JFK for flights departing between 3:00 and 4:00 PM on Wednesday, is uniformly distributed between 8 and 16 minutes.

Use the Distributions tool to help you answer the questions that follow.

0123NormalStandard NormalUniform

Select a Distribution

The height of the graph of the probability density function f(x) varies with X as follows (round to four decimal places):

X	Height of the Graph of the Probability Density Function
X < 8
8 ≤ X ≤ 16
X > 16

You are flying out of Terminal 3 at JFK on a Wednesday afternoon between 3:00 and 4:00 PM. You get stuck in a traffic jam on the way to the airport, and if it takes you longer than 12 minutes to clear security, you’ll miss your flight. The probability that you'll miss your flight is-------------------- .

You have arrived at the airport and have been waiting 10 minutes at the security checkpoint. Recall that if you spend more than 12 minutes clearing security, you will miss your flight. Now what is the probability that you'll miss your flight?

a. 0.25

b. 0.6667

c. 0.5

d. 0.8333

An automobile battery manufacturer offers a 39/50 warranty on its batteries. The first number in the warranty code is the free-replacement period; the second number is the prorated-credit period. Under this warranty, if a battery fails within 39 months of purchase, the manufacturer replaces the battery at no charge to the consumer. If the battery fails after 39 months but within 50 months, the manufacturer provides a prorated credit toward the purchase of a new battery.

The manufacturer assumes that X, the lifetime of its auto batteries, is normally distributed with a mean of 44 months and a standard deviation of 3.6 months.

Use the following Distributions tool to help you answer the questions that follow. (Hint: When you adjust the parameters of a distribution, you must reposition the vertical line (or lines) for the correct areas to be displayed.)

0123BinomialChi-SquareExponentialF DistributionHypergeometricNormalUniform

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If the manufacturer’s assumptions are correct, it would need to replace------------------------of its batteries free of charge.

The company finds that it is replacing 9.34% of its batteries free of charge. It suspects that its assumption about the standard deviation of the life of its batteries is incorrect. A standard deviation of---------------results in a 9.34% replacement rate.

Using the revised standard deviation for battery life, what percentage of the manufacturer’s batteries don’t qualify for free replacement but do qualify for the prorated credit?

a. 84.95%

b.44.29%

c. 40.66%

d. 5.71%

Answer 1

1.

(A)

The distribution which X follows is uniform distribution.

(B)

(C)

Required probability is given by

(D)

Required probability is given by

So, (b) 0.6667.

2.

(A)

The distribution which X follows is normal distribution.

(B)

Probability that a battery fails within 39 months is given by

   [Using R-code 'pnorm(-1.388889)']

So, the company need to replace 8.24% of the batteries free of charge.

(C)

Suppose, standard deviation be k.

   [Using R-code 'qnorm(0.0934)']

So, a standard deviation of 3.787578 results in a 9.34% replacement rate.

(D)

Using the revised standard deviation for battery life, probability that a battery qualifies for prorated credit is given by

Using R-code 'pnorm(1.58)-pnorm(-1.32)']

Hence, (a) 84.95% of manufacturer’s batteries don’t qualify for free replacement but qualify for the prorated credit.