/*! 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 37 Let \(x\) denote the time taken ... [FREE SOLUTION] | 91影视

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Chapter 6: Problem 37

Let \(x\) denote the time taken to run a road race. Suppose \(x\) is approximately normally distributed with a mean of 190 minutes and a standard deviation of 21 minutes. If one runner is selected at random, what is the probability that this runner will complete this road race a. in less than 160 minutes? b. in 215 to 245 minutes?

Short Answer

Expert verified
a. The probability that the runner will complete this road race in less than 160 minutes is approximately 7.64%. b. The probability that the runner will complete this road race in 215 to 245 minutes is about 11.36%.

Step by step solution

01

- Calculate Z-score for 160 minutes

The Z-score is calculated by subtracting the mean from the point in question and then dividing by the standard deviation. So, for 160 minutes, the calculation would be \(Z = (160-190)/21 = -1.43\)
02

- Find the probability for finishing in less than 160 minutes

From a Z-table or using a calculator, one can get the left-tail probability value for Z = -1.43. The probability is about 0.0764, meaning there is approximately a 7.64% chance a runner will complete the race in less than 160 minutes.
03

- Calculate Z-scores for 215 and 245 minutes

Similarly calculate the Z-scores for 215 minutes and 245 minutes. \(Z1 = (215-190)/21 = 1.19\) and \(Z2 = (245-190)/21 = 2.62\)
04

- Find the probability of finishing between 215 to 245 minutes

Look up the corresponding probabilities for Z1 and Z2 in the table or using a calculator. The probabilities are 0.8820 and 0.9956, respectively. To find the probability of a time between 215 and 245 minutes, subtract the probability for Z1 from the probability for Z2: 0.9956 - 0.8820 = 0.1136. This indicates roughly an 11.36% chance a runner will complete the race in between 215 and 245 minutes.

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