/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 41 A construction zone on a highway... [FREE SOLUTION] | 91影视

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Chapter 6: Problem 41

A construction zone on a highway has a posted speed limit of 40 miles per hour. The speeds of vehicles passing through this construction zone are normally distributed with a mean of 46 miles per hour and a standard deviation of 4 miles per hour. Find the percentage of vehicles passing through this construction zone that are a. exceeding the posted speed limit b. traveling at speeds between 50 and 57 miles per hour

Short Answer

Expert verified
a. 93.32% of vehicles are exceeding the posted speed limit.\nb. 15.57% of vehicles are traveling at speeds between 50 and 57 miles per hour.

Step by step solution

01

Interpret the given mean and standard deviation

The mean is 46 miles per hour, this is the average speed of the vehicles passing through the construction zone. The standard deviation is 4 miles per hour, indicating the typical variation or spread of the speeds around the mean.
02

Calculate the z-score for the posted speed limit

The z-score represents how many standard deviations a value is from the mean. The formula to calculate the z-score is \( Z = (X - \mu) / \sigma \). Here, \( X = 40 \) is the posted speed limit, \( \mu = 46 \) is the mean and \( \sigma = 4 \) is the standard deviation. So, the z-score for the speed limit is \( Z = (40 - 46) / 4 = -1.5 \).
03

Find the percentage of vehicles exceeding the speed limit

Using standard normal distribution tables, the probability associated with \( Z = -1.5 \) is 0.0668. Because the question asks for the percentage of vehicles exceeding the speed limit, we need to find the upper tail of the distribution, which is \( 1 - 0.0668 = 0.9332 \) or 93.32%.
04

Calculate the z-scores for the speeds 50 and 57 miles per hour

Using the same formula for the z-score, we get \( Z = (50 - 46) / 4 = 1 \) for 50 miles per hour and \( Z = (57 - 46) / 4 = 2.75 \) for 57 miles per hour.
05

Find the percentage of vehicles traveling at speeds between 50 and 57 miles per hour

Using the standard normal distribution tables, the probabilities for these z-scores are 0.8413 for \( Z = 1 \) and 0.9970 for \( Z = 2.75 \). To find the percentage of vehicles traveling between these speeds, subtract these probabilities: \( 0.9970 - 0.8413 = 0.1557 \), or 15.57%.

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Most popular questions from this chapter

The amount of time taken by a bank teller to serve a randomly selected customer has a normal distribution with a mean of 2 minutes and a standard deviation of \(.5\) minute. a. What is the probability that both of two randomly selected customers will take less than I minute each to be served? b. What is the probability that at least one of four randomly selected customers will need more than \(2.25\) minutes to be served?

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