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Stat.ch6and7

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Statistics
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University of California Irvine
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Homework
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Chapter 6:
5. In October 2012, Apple introduced a much smaller variant of the Apple iPad, known as the
iPad Mini. Weighing less than 11 ounces, it was about 50% lighter than the standard iPad.
Battery tests for the iPad Mini showed a mean life of 10.25 hours (The Wall Street Journal,
October 31, 2012). Assume the battery life of the iPad Mini is uniformly distributed between
8.5 and 12 hours.
a. Give a mathematical expression for the probability density function of battery life.
The uniform distribution is between 8.5 and 12 hours, we take the difference,
12-8.5=3.5
Therefore, the probability density function is,


 
b. What is the probability that the battery life for an iPad Mini will be 10 hours or less?
The cumulative distribution function is,


Therefore,

  


The probability that the battery life will be 10 hours or less is 0.4286.
c. What is the probability that the battery life for an iPad Mini will be at least 11 hours?
We want to determine,
 which is equivalent to   

 



The probability that the battery life is at least 11 hours is 0.2857.
d. What is the probability that the battery life for an iPad Mini will be between 9.5 and
11.5 hours?
We want to determine,
 which is equivalent to

 
This is,
  

  


The probability that the battery life is between 9.5 and 11.5 hours is 0.5714.
e. In a shipment of 100 iPad Minis, how many should have a battery life of at least 9
hours?
First, we determine the probability that the batter life is at least 9 hours.
  
 
  


If we have 100 mini iPad Minis, 
About 86 iPad Minis should have a batter life of at least 9 hours.

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23. The time needed to complete a final examination in a particular college course is normally
distributed with a mean of 80 minutes and a standard deviation of 10 minutes. Answer the
following questions:
a. What is the probability of completing the exam in one hour or less?
One hour is 60 minutes
We standardize 60 minutes.
 


We want to determine , using a z-table, we know


Therefore, the probability of completing the exam in one hour or less is 0.0228.
b. What is the probability that a student will complete the exam in more than 60 minutes
but less than 75 minutes?
With X being the time it takes to complete the exam, we want to determine:

We standardize,
 

 


Which is equal to:

 

 
Therefore, the probability of completing the exam in more than 60 minutes but less
than 75 minutes is 0.2857.
c. Assume that the class has 60 students and that the examination period is 90 minutes in
length. How many students do you expect will be unable to complete the exam in the
allotted time?
First, we determine the probability of completing the exam in more than 90 minutes.

 


Which is equal to   
 
Therefore, if we have 60 students,




We expect 10 students to be unable to complete the exam withing the allotted time.
37. Wendy’s restaurant has been recognized for having the fastest average service time among
fast food restaurants. In a benchmark study, Wendy’s average service time of 2.2 minutes
was less than those of Burger King, Chick-fil-A, Krystal, McDonald’s, Taco Bell, and Taco
John’s (QSR Magazine website, December 2014). Assume that the service time for Wendy’s
has an exponential distribution.
a. What is the probability that a service time is less than or equal to one minute?
Since the average service time is 2.2, we have a parameter of


Which gives es the probability density function:




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Chapter 6: 5. In October 2012, Apple introduced a much smaller variant of the Apple iPad, known as the iPad Mini. Weighing less than 11 ounces, it was about 50% lighter than the standard iPad. Battery tests for the iPad Mini showed a mean life of 10.25 hours (The Wall Street Journal, October 31, 2012). Assume the battery life of the iPad Mini is uniformly distributed between 8.5 and 12 hours. a. Give a mathematical expression for the probability density function of battery life. The uniform distribution is between 8.5 and 12 hours, we take the difference, 12-8.5=3.5 Therefore, the probability density function is, 1 8.5 ≤ 𝑥 ≤ 12 𝑓(𝑥) = {3.5 , 0, 𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒 b. What is the probability that the battery life for an iPad Mini will be 10 hours or less? The cumulative distribution function is, 𝑥−8.5 3.5 Therefore, 10 − 8.5 = 0.4286 3.5 The probability that the battery life will be 10 hours or less is 0.4286. 𝑃(𝑋 ≤ 10) = c. What is the probability that the battery life for an iPad Mini will be at least 11 hours? We want to determine, 𝑃(𝑋 ≥ 11) which is equivalent to 1 − 𝑃(𝑋 < 11) = 1 − 11−8.5 3.5 = 0.2857 The probability that the battery life is at least 11 hours is 0.2857. d. What is the probability that the battery life for an iPad Mini will be between 9.5 and 11.5 hours? We want to determine, 𝑃(9.5 ≤ 𝑋 ≤ 11.5) which is equivalent to 𝑃(𝑋 < 11.5) − 𝑃(𝑋 < 9.5) This is, 11.5 − 8.5 ...
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