1. Suppose a manufacturer (Mary) wants to guess how many units a competitor (John) completes per week. Since she cannot access John’s factory and records but can buy the competitor’s goods on the open market she decides to buy 14 random units. She then sends you the following serial numbers of the units:
54

33

72

48

21

42

24

75

57

29

59

37

48

55

 

a. Can you find a way of guesstimating the number of units the competitor produces per week?​​​​​​​[30 marks]
CLUE: This a probability question

This problem involves estimating the total number of units a competitor produces per week using the given serial numbers of purchased units. The approach we can use here is based on the *German tank problem*, a statistical method used during World War II to estimate the total number of enemy tanks produced, based on serial numbers of captured or observed tanks.

### Estimating the Total Number of Units Produced

Given serial numbers:
54, 33, 72, 48, 21, 42, 24, 75, 57, 29, 59, 37, 48, 55

Let's denote:
- \( N \) as the total number of units produced by the competitor.
- \( k \) as the number of serial numbers observed (14 in this case).
- \( m \) as the maximum observed serial number (which is 75 in this case).

The maximum likelihood estimator (MLE) for \( N \) is given by:

\[
\hat{N} = m \times \left(1 + \frac{1}{k}\right) - 1
\]

Where:
- \( m = 75 \)
- \( k = 14 \)

Let's calculate this:

\[
\hat{N} = 75 \times \left(1 + \frac{1}{14}\right) - 1
\]

\[
\hat{N} = 75 \times \left(1 + 0.0714\right) - 1
\]

\[
\hat{N} = 75 \times 1.0714 - 1
\]

\[
\hat{N} = 80.355 - 1 = 79.355
\]

Rounding to the nearest whole number:

\[
\hat{N} \approx 79
\]

### Conclusion

The estimated total number of units that the competitor (John) produces per week is approximately **79**.

### Explanation of the Approach

- **German Tank Problem**: This method was used by the Allies during WWII to estimate the number of German tanks by analyzing serial numbers on captured tanks.
- **Maximum Likelihood Estimation**: This is a statistical method used to estimate parameters of a probability distribution by maximizing a likelihood function.
- **Serial Numbers**: By using the maximum observed serial number and the number of observed serial numbers, we estimate the total production.

This estimate assumes that the serial numbers are sequential and that every number from 1 to \( N \) has an equal chance of being observed.

---

Would you like more details on any part of this solution or have any other questions?

Here are five related questions to deepen your understanding:

1. What assumptions does the German tank problem make, and how do they apply in this scenario?
2. How would the estimate change if the sample size (the number of serial numbers observed) were larger?
3. How does the maximum serial number observed influence the estimate of total production?
4. What are the limitations of using the maximum likelihood estimator in this context?
5. How can the variance of this estimator be calculated?

**Tip:** When dealing with estimation problems, understanding the underlying assumptions is key to evaluating the accuracy and reliability of the results.

Suppose a manufacturer (Mary) wants to guess how many units a competitor (John) completes per week. Since she cannot access John’s factory and records but can buy the competitor’s goods on the open market, she decides to buy 14 random units. She then sends you the following serial numbers of the units: 54, 33, 72, 48, 21, 42, 24, 75, 57, 29, 59, 37, 48, 55. Can you find a way of guesstimating the number of units the competitor produces per week?

Learn how to estimate the total production units of a competitor using serial numbers of purchased goods. This problem applies the German Tank Problem and Maximum Likelihood Estimation, making it suitable for undergraduate-level probability courses.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation