91Ó°ÊÓ

A sample of \(n\) captured Pandemonium jet fighters results in serial numbers \(x_{1}, x_{2}, x_{3}, \ldots, x_{n}\). The CIA knows that the aircraft were numbered consecutively at the factory starting with \(\alpha\) and ending with \(\beta\), so that the total number of planes manufacture is \(\beta-\alpha+1\) (e.g., if \(\alpha=17\) and \(\beta=29\), then \(29-17+1=13\) planes having serial numbers \(17,18,19, \ldots, 28,29\) were manufactured). However, the CIA does not know the values of \(\alpha\) or \(\beta\). A CIA statistician suggests using the esti\(\operatorname{mator} \max \left(X_{i}\right)-\min \left(X_{i}\right)+1\) to estimate the total number of planes manufactured. a. If \(n=5, x_{1}=237, x_{2}=375, x_{3}=202\), \(x_{4}=525\), and \(x_{5}=418\), what is the corresponding estimate? b. Under what conditions on the sample will the value of the estimate be exactly equal to the true total number of planes? Will the estimate ever be larger than the true total? Do you think the estimator is unbiased for estimating \(\beta-\) \(\alpha+1\) ? Explain in one or two sentences. (A similar method was used to estimate German tank production in World War II.)

Step by step solution

01

Identify Given Values

From the problem, we have a sample of \( n = 5 \) serial numbers: \( x_1 = 237 \), \( x_2 = 375 \), \( x_3 = 202 \), \( x_4 = 525 \), \( x_5 = 418 \). These numbers represent the captured serial numbers of the Pandemonium jet fighters.

02

Calculate Maximum and Minimum of Sample

Find the maximum and minimum serial numbers from the sample. Here, the maximum serial number, \( \max(X_i) \), is 525, and the minimum serial number, \( \min(X_i) \), is 202.

03

Use the Estimator Formula

Substitute the maximum and minimum values into the estimator formula to estimate the total number of planes: \[ \max(X_i) - \min(X_i) + 1 = 525 - 202 + 1 = 324. \] So, the estimate suggests that 324 planes were manufactured.

04

Conditions for Accuracy

For the estimator to be exactly equal to the true total number of planes, the sample must include both serial numbers \( \alpha \) (the first plane) and \( \beta \) (the last plane).

05

Discuss the Estimator Properties

The estimator will never exceed the true total number of planes. However, it could potentially underestimate if serial numbers outside the sample range were manufactured. This suggests that the estimator is likely a biased estimator for the true total \( \beta - \alpha + 1 \), as it tends to underestimate unless the sample includes both extreme serial numbers.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Estimation Methods

In statistics, estimation methods are techniques used to infer the properties of a population based on a sample. The goal is to draw conclusions about the entire population while only examining a part of it. There is a variety of estimation methods, but broadly, they fall into two categories: point estimation and interval estimation.

• Point Estimation: This provides a single value as an estimate of the population parameter. In the exercise, the estimator used is a point estimate, calculated as the difference between the maximum and minimum serial numbers in the sample plus one.
• Interval Estimation: This method gives a range of values such that the parameter is believed to lie within that range with a certain degree of confidence.

Estimation methods are chosen based on the problem context and the desired precision. In our example with Pandemonium jet fighters, the chosen point estimation method is simple and straightforward, based on the maximum and minimum values.

Serial Number Analysis

Serial number analysis is a technique used to estimate the production quantity of items when the items have identifiable serial numbers. This method was famously used during World War II to estimate the production of German tanks.

By analyzing the serial numbers of captured equipment, one can estimate how many items were produced between the lowest and highest numbers observed. The formula \( \max(X_i) - \min(X_i) + 1 \) is applied here to estimate the number of planes manufactured.

Serial number analysis is particularly useful when:

• Items are marked sequentially.
• Only a sample of the total production is available.
• The sequential numbering captures both the start and end numbers.

This method relies heavily on having complete sequences and can be skewed if partially sampled.

Sampling Theory

Sampling theory is a scientific method of drawing insights about a population based on samples. It involves principles that guide how samples are collected and analyzed to infer the population's characteristics.

Key elements of sampling theory include:

• Random Sampling: Each member of the population has an equal chance of being selected, helping to ensure unbiased samples.
• Sample Size: Larger samples tend to provide more accurate estimates, as they are likely to represent the population better.
• Sampling Error: This is the discrepancy between the sample estimate and the true population parameter, often reduced by increasing the sample size.

In the context of the Pandemonium jet fighters, sampling theory clarifies that using more serial numbers (hence larger samples) can potentially increase the accuracy of the number of planes estimated.

Estimator Bias

Estimator bias is a key concept in statistics—it's the difference between an estimator's expected value and the true value of the parameter being estimated. An unbiased estimator has an expected value equal to the true parameter. In the Pandemonium jet fighter exercise, the estimator used (\( \max(X_i) - \min(X_i) + 1 \)) might be biased. This means it could systematically overestimate or, more likely, underestimate the total number of planes unless the sample contains both the lowest and highest serial numbers produced. This bias arises because:

• The sample may not fully capture the range of serial numbers.
• Serial numbers outside the sampled range might exist, leading to underestimation.

Thus, while simple, this estimation method may not be ideal for accurately determining production quantity unless conditions align perfectly with the assumptions.