Answer :
Let's break down each part of this problem step-by-step.
### a. Mathematical Expression for the Probability Density Function
Given are three options for the probability density function [tex]\( f(x) \)[/tex] of the battery life:
- Option A:
[tex]\[ f(x) = \left\{ \begin{array}{cc} \frac{1}{3.5} & \text{for } 8.5 \leq x \leq 12 \\ 0 & \text{elsewhere} \end{array} \right. \][/tex]
- Option B:
[tex]\[ f(x) = \left\{ \begin{array}{cc} \frac{2}{3.5} & \text{for } 8.5 \leq x \leq 12 \\ 0 & \text{elsewhere} \end{array} \right. \][/tex]
- Option C:
[tex]\[ f(x) = \left\{ \begin{array}{cc} \frac{1}{1.5} & \text{for } 8.5 \leq x \leq 10 \\ 0 & \text{elsewhere} \end{array} \right. \][/tex]
The correct mathematical expression is:
[tex]\[ A \][/tex]
### b. Probability that Battery Life Will Be 10 Hours or Less
The probability that the battery life is 10 hours or less can be found by integrating the probability density function [tex]\( f(x) \)[/tex] from 8.5 to 10:
[tex]\[ P(X \leq 10) = \int_{8.5}^{10} f(x) \, dx \][/tex]
Given [tex]\( f(x) = \frac{1}{3.5} \)[/tex] over this interval:
[tex]\[ P(X \leq 10) = (10 - 8.5) \cdot \frac{1}{3.5} = \frac{1.5}{3.5} \approx 0.4286 \][/tex]
So, the probability is:
[tex]\[ 0.4286 \][/tex]
### c. Probability that Battery Life Will Be at Least 11 Hours
The probability that the battery life is at least 11 hours can be found by integrating [tex]\( f(x) \)[/tex] from 11 to 12:
[tex]\[ P(X \geq 11) = \int_{11}^{12} f(x) \, dx \][/tex]
Given [tex]\( f(x) = \frac{1}{3.5} \)[/tex] over this interval:
[tex]\[ P(X \geq 11) = (12 - 11) \cdot \frac{1}{3.5} = \frac{1}{3.5} \approx 0.2857 \][/tex]
So, the probability is:
[tex]\[ 0.2857 \][/tex]
### d. Probability that Battery Life Will Be Between 9.5 and 11.5 Hours
The probability that the battery life is between 9.5 and 11.5 hours can be found by integrating [tex]\( f(x) \)[/tex] from 9.5 to 11.5:
[tex]\[ P(9.5 \leq X \leq 11.5) = \int_{9.5}^{11.5} f(x) \, dx \][/tex]
Given [tex]\( f(x) = \frac{1}{3.5} \)[/tex] over this interval:
[tex]\[ P(9.5 \leq X \leq 11.5) = (11.5 - 9.5) \cdot \frac{1}{3.5} = \frac{2}{3.5} \approx 0.5714 \][/tex]
So, the probability is:
[tex]\[ 0.5714 \][/tex]
### e. Number of iPad Minis with Battery Life of At Least 9 Hours
To find the number of iPad Minis with a battery life of at least 9 hours in a shipment of 100 units, we compute the probability that the battery life is at least 9 hours and then multiply it by 100:
[tex]\[ P(X \geq 9) = \int_{9}^{12} f(x) \, dx \][/tex]
Given [tex]\( f(x) = \frac{1}{3.5} \)[/tex] over this interval:
[tex]\[ P(X \geq 9) = (12 - 9) \cdot \frac{1}{3.5} = \frac{3}{3.5} \][/tex]
Multiplying by 100:
[tex]\[ \text{Number of iPad Minis} = \frac{3}{3.5} \cdot 100 \approx 85.7143 \][/tex]
Rounded to the nearest whole number:
[tex]\[ 86 \][/tex]
So, the number of iPad Minis with at least 9 hours of battery life is:
[tex]\[ 86 \][/tex]
### a. Mathematical Expression for the Probability Density Function
Given are three options for the probability density function [tex]\( f(x) \)[/tex] of the battery life:
- Option A:
[tex]\[ f(x) = \left\{ \begin{array}{cc} \frac{1}{3.5} & \text{for } 8.5 \leq x \leq 12 \\ 0 & \text{elsewhere} \end{array} \right. \][/tex]
- Option B:
[tex]\[ f(x) = \left\{ \begin{array}{cc} \frac{2}{3.5} & \text{for } 8.5 \leq x \leq 12 \\ 0 & \text{elsewhere} \end{array} \right. \][/tex]
- Option C:
[tex]\[ f(x) = \left\{ \begin{array}{cc} \frac{1}{1.5} & \text{for } 8.5 \leq x \leq 10 \\ 0 & \text{elsewhere} \end{array} \right. \][/tex]
The correct mathematical expression is:
[tex]\[ A \][/tex]
### b. Probability that Battery Life Will Be 10 Hours or Less
The probability that the battery life is 10 hours or less can be found by integrating the probability density function [tex]\( f(x) \)[/tex] from 8.5 to 10:
[tex]\[ P(X \leq 10) = \int_{8.5}^{10} f(x) \, dx \][/tex]
Given [tex]\( f(x) = \frac{1}{3.5} \)[/tex] over this interval:
[tex]\[ P(X \leq 10) = (10 - 8.5) \cdot \frac{1}{3.5} = \frac{1.5}{3.5} \approx 0.4286 \][/tex]
So, the probability is:
[tex]\[ 0.4286 \][/tex]
### c. Probability that Battery Life Will Be at Least 11 Hours
The probability that the battery life is at least 11 hours can be found by integrating [tex]\( f(x) \)[/tex] from 11 to 12:
[tex]\[ P(X \geq 11) = \int_{11}^{12} f(x) \, dx \][/tex]
Given [tex]\( f(x) = \frac{1}{3.5} \)[/tex] over this interval:
[tex]\[ P(X \geq 11) = (12 - 11) \cdot \frac{1}{3.5} = \frac{1}{3.5} \approx 0.2857 \][/tex]
So, the probability is:
[tex]\[ 0.2857 \][/tex]
### d. Probability that Battery Life Will Be Between 9.5 and 11.5 Hours
The probability that the battery life is between 9.5 and 11.5 hours can be found by integrating [tex]\( f(x) \)[/tex] from 9.5 to 11.5:
[tex]\[ P(9.5 \leq X \leq 11.5) = \int_{9.5}^{11.5} f(x) \, dx \][/tex]
Given [tex]\( f(x) = \frac{1}{3.5} \)[/tex] over this interval:
[tex]\[ P(9.5 \leq X \leq 11.5) = (11.5 - 9.5) \cdot \frac{1}{3.5} = \frac{2}{3.5} \approx 0.5714 \][/tex]
So, the probability is:
[tex]\[ 0.5714 \][/tex]
### e. Number of iPad Minis with Battery Life of At Least 9 Hours
To find the number of iPad Minis with a battery life of at least 9 hours in a shipment of 100 units, we compute the probability that the battery life is at least 9 hours and then multiply it by 100:
[tex]\[ P(X \geq 9) = \int_{9}^{12} f(x) \, dx \][/tex]
Given [tex]\( f(x) = \frac{1}{3.5} \)[/tex] over this interval:
[tex]\[ P(X \geq 9) = (12 - 9) \cdot \frac{1}{3.5} = \frac{3}{3.5} \][/tex]
Multiplying by 100:
[tex]\[ \text{Number of iPad Minis} = \frac{3}{3.5} \cdot 100 \approx 85.7143 \][/tex]
Rounded to the nearest whole number:
[tex]\[ 86 \][/tex]
So, the number of iPad Minis with at least 9 hours of battery life is:
[tex]\[ 86 \][/tex]
