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Helen [10]
2 years ago
14

The travel time on a section of a Long Island Expressway (LIE) is normally distributed with a mean of 80 seconds and a standard

deviation of 6 seconds. What travel time separates the top 2.5% of the travel times from the rest
Mathematics
1 answer:
user100 [1]2 years ago
3 0

Answer:

The travel time that separates the top 2.5% of the travel times from the rest is of 91.76 seconds.

Step-by-step explanation:

Normal Probability Distribution

Problems of normal distributions can be solved using the z-score formula.

In a set with mean \mu and standard deviation \sigma, the z-score of a measure X is given by:

Z = \frac{X - \mu}{\sigma}

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the p-value, we get the probability that the value of the measure is greater than X.

Mean of 80 seconds and a standard deviation of 6 seconds.

This means that \mu = 80, \sigma = 6

What travel time separates the top 2.5% of the travel times from the rest?

This is the 100 - 2.5 = 97.5th percentile, which is X when Z has a p-value of 0.975, so X when Z = 1.96.

Z = \frac{X - \mu}{\sigma}

1.96 = \frac{X - 80}{6}

X - 80 = 6*1.96

X = 91.76

The travel time that separates the top 2.5% of the travel times from the rest is of 91.76 seconds.

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