We observe 14 components, starting from time t = 0. The component life times are independent. Observation ends immediately following the 4th failure, which occurs at time t = 13.48. Give (correct to 2 decimal places) an appropriate estimate for F(13.48), using the Mean ranking method.

To estimate \( F(13.48) \) using the Mean Ranking Method, we can follow these steps:

### 1. Understanding the Problem
- We have 14 components under observation.
- Observation ends immediately after the 4th failure at time \( t = 13.48 \).
- \( F(t) \) is the cumulative distribution function, which gives the probability that a component will have failed by time \( t \).

### 2. Rank the Failures
Since the observation ends after the 4th failure at time \( t = 13.48 \), we rank the failure times as follows:
- The 1st failure time \( t_1 \)
- The 2nd failure time \( t_2 \)
- The 3rd failure time \( t_3 \)
- The 4th failure time \( t_4 = 13.48 \)

### 3. Calculate the Estimate for \( F(13.48) \)
The Mean Ranking Method estimate for \( F(t) \) is given by:
\[
F(t) = \frac{\text{Rank of failure at } t}{\text{Total number of components + 1}}
\]
For \( t = 13.48 \) (which is the 4th failure), the rank is 4.

So, the estimate is:
\[
F(13.48) = \frac{4}{14 + 1} = \frac{4}{15} \approx 0.27
\]

### Final Answer:
An appropriate estimate for \( F(13.48) \) using the Mean Ranking Method is **0.27**.

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Would you like further details or have any questions?

Here are five related questions to expand your understanding:

1. How would the estimate change if the observation ended at the 5th failure instead of the 4th?
2. What is the significance of using the Mean Ranking Method in reliability analysis?
3. How does the number of components under observation affect the accuracy of the estimate?
4. What other methods can be used to estimate \( F(t) \) in reliability analysis?
5. How would you interpret the result \( F(13.48) = 0.27 \) in a practical scenario?

**Tip:** Understanding the assumptions behind the method used (e.g., independence of component lifetimes) is crucial for correctly applying statistical methods in reliability analysis.

Learn how to estimate F(13.48) using the Mean Ranking Method in reliability analysis. Understand the calculation steps and assumptions involved, with a practical example involving 14 components and their failure times. Suitable for advanced undergraduate students and professionals in reliability engineering.

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