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Chapter 6: Q. 6.31 (page 260)

Waiting for the Train. A commuter train arrives punctually at a station every half hour. Each morning, a commuter named John leaves his house and casually strolls to the train station. The time, in minutes, that John waits for a train is a variable with density curve y=130for0<x<30,y=0and otherwise.

(a) Graph the density curve of this variable.

(b) Show that the area under this density curve to the left of any number x between 0 and 30equals x30.

What percentage of all possible observations of the variable

(c) less than 5 minutes.

(d) between 10and 15 minutes.

(e) atleast20minutes.

Short Answer

Expert verified

(a) The graph of the density function is:

(b) Thus, the area under this density curve to the left of any number x between 0 and 30equals x30.

(c) John waits 16.67% of time less than 5 minutes for a train.

(d) John waits 16.67% of time between 10, 15minutes of the train.

(e) John waits 33.33%of time at least 20 minutes of the train.

Step by step solution

01

Part (a) Step 1. Given Information. 

A variable has a density curve whose equation is y=130for0<x<30,y=0.

02

Part (a) Step 2. Graph the density function. 

Graph the density function y=130for0<x<30,y=0.

03

Part (b) Step 1. The area between 0, 30. 

From part (a),

Base,b=x

Height, h=130

Thus the area of the rectangle is given by,

A=bh=x(130)=x30

04

Part (c) Step 1. Find the percentage. 

Area to the left of 5=530

role="math" localid="1652600779734" =0.1667

John waits 16.67% time of less than 5 minutes for a train.

05

Part (d) Step 1. Find the percentage. 

Area between 10,15=[Area to the left of 15]-[Area to the left of 10]

role="math" localid="1652601050742" =1530-1030=530=0.1667

John waits 16.67% of time between 10, 15 minutes of the train.

06

Part (e) Step 1. Find the percentage. 

[Area to the right of 20]=1-[Area to the left of 20]

=1-2030=1030=0.3333

John waits 33.33% of time at least20 minutes of the train.

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