The average lifetime of a certain new cell phone is 3.4 years. The manufacturer will replace any cell phone failing within three years of the date of purchase. The lifetime of these cell phones is known to follow an exponential distribution. 68% of these phone last how long (in years)?

To find the range of lifetimes within which 68% of the phones will last, given that their lifetimes follow an exponential distribution with a mean of 3.4 years, we'll follow these steps:

### Step 1: Key Parameters
The lifetime \( T \) of these cell phones follows an exponential distribution with a mean \( \mu = 3.4 \) years. The rate parameter \( \lambda \) is given by:

\[
\lambda = \frac{1}{\mu} = \frac{1}{3.4} \approx 0.2941
\]

### Step 2: Exponential Distribution Properties
For an exponential distribution, the standard deviation is equal to the mean \( \mu \). In a normal distribution, 68% of the data lies within one standard deviation of the mean. However, exponential distributions are not symmetric like normal distributions, so we need to directly compute the range.

### Step 3: Solving for Percentiles
We previously calculated that the range covering 68% of the lifetimes falls between the 16th and 84th percentiles of the exponential distribution. This corresponds to the range of:

- **16th percentile**: \( \approx 0.59 \) years
- **84th percentile**: \( \approx 6.23 \) years

### Final Answer:
68% of the cell phones are expected to last between **0.59 years** and **6.23 years**.

If you have any more questions or need further clarification, feel free to ask!

### Related Questions:
1. What is the probability that a phone will last exactly 3 years?
2. How do you calculate the expected number of replacements based on the manufacturer’s policy?
3. What is the relationship between the exponential distribution and the Poisson distribution?
4. How does changing the mean lifetime affect the reliability of the phones?
5. What is the median lifetime of these cell phones?

### Tip:
For exponential distributions, the probability of an event occurring after a certain time decreases exponentially, making it useful for modeling processes like lifetimes and waiting times.

Learn how to calculate the range covering 68% of the lifetime of cell phones following an exponential distribution with a mean of 3.4 years. Understand the properties of exponential distributions and how to compute percentiles. Suitable for advanced high school students and beyond.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation