/*! 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 27 A construction zone on a highway... [FREE SOLUTION] | 91影视

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Chapter 6: Problem 27

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. The percentage of vehicles exceeding the posted speed limit of 40 mph is 93.32%. b. The percentage of vehicles traveling at speeds between 50 and 57 mph is 15.57%.

Step by step solution

01

Calculate z-score for the posted speed limit

To find the percentage of vehicles that exceed the speed limit, calculate the z-score using the formula: \( z = \frac{(X - 渭)}{蟽} \), where X is the posted speed limit (40 mph), 渭 is the mean (46 mph), and 蟽 is the standard deviation (4 mph). Therefore, \( z = \frac{(40 - 46)}{4} = -1.5 \).
02

Find the percentage for this area from z-table

By referring to the z-table or using a online calculator, the percentage associated with z = -1.5 is 0.0668 or 6.68%. However, since we are interested in the vehicles which are exceeding the speed limit, we look at the area to the right of the z score, which gives the percentage as 1 - 0.0668 = 0.9332 or 93.32%.
03

Calculate z-scores for speeds between 50 and 57 mph

Calculate the z-scores for 50 mph and 57 mph respectively using the z-score formula. This gives \( z_1 = \frac{(50 - 46)}{4} = 1 \) and \( z_2 = \frac{(57 - 46)}{4} = 2.75 \).
04

Find the percentage for this range from z-table

By referring to the z-table or using a online calculator, the percentages associated with \( z_1 = 1 \) is 0.3413 and \( z_2 = 2.75 \) is 0.4970. The percentage of vehicles traveling at speeds between 50 and 57 mph is \( 0.4970 - 0.3413 = 0.1557 or 15.57% \).

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