91Ó°ÊÓ

As part of a quality control process for computer chips, an engineer at a factory randomly samples 212 chips during a week of production to test the current rate of chips with severe defects. She finds that 27 of the chips are defective. (a) What population is under consideration in the data set? (b) What parameter is being estimated? (c) What is the point estimate for the parameter? (d) What is the name of the statistic can we use to measure the uncertainty of the point estimate? (e) Compute the value from part (d) for this context. (f) The historical rate of defects is \(10 \%\). Should the engineer be surprised by the observed rate of defects during the current week? (g) Suppose the true population value was found to be \(10 \%\). If we use this proportion to recompute the value in part (e) using \(p=0.1\) instead of \(\hat{p},\) does the resulting value change much?

Step by step solution

01

Identify the Population

The population under consideration is all the computer chips produced in the factory during the week of production. This includes every chip made in that time frame, not just the ones sampled.

02

Determine the Parameter

The parameter being estimated is the true proportion of defective chips in the entire population of chips produced during the week. It's a value that represents the overall defect rate for all chips made in that period.

03

Calculate the Point Estimate

The point estimate for the proportion of defective chips is calculated from the sample. Since 27 out of 212 chips are defective, the point estimate \( \hat{p} \) is: \[ \hat{p} = \frac{27}{212} \approx 0.127 \]

04

Identify the Statistic for Uncertainty

The statistic used to measure the uncertainty of the point estimate is the Standard Error (SE). It provides an estimate of the variability of the sample proportion around the true population proportion.

05

Compute the Standard Error

The standard error for a proportion is calculated as: \[ SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} \] where \( \hat{p} = 0.127 \) and \( n = 212 \). Substituting the values, we compute: \[ SE = \sqrt{\frac{0.127 \times (1 - 0.127)}{212}} \approx 0.0242 \]

06

Compare with Historical Defect Rate

The historical defect rate is 10%. To decide if the engineer should be surprised, check if the observed defect rate is within a reasonable range given the historical rate. The observed rate is 12.7%, which is close to 10%, but a hypothesis test or confidence interval might provide more insight.

07

Recompute with Assumed Population Proportion

If we use the assumed population proportion \( p = 0.1 \) to recompute the standard error, the formula becomes: \[ SE = \sqrt{\frac{0.1 \times (1 - 0.1)}{212}} \approx 0.0209 \]. This is similar to the standard error computed with \( \hat{p} = 0.127 \).

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Key Concepts

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

Population Parameter

In the context of statistical inference, the population parameter is a critical concept. It represents a value or measurement that we aim to estimate from a population.

In our exercise involving computer chips, the population parameter in question is the true proportion of defective chips produced. We seek to understand the defect rate in every chip manufactured during that week.

Essentially, the population parameter offers a comprehensive overview of a particular attribute in the entire population, not just a sample. By understanding this, we gain a full picture of the quality over the entire production period, which is essential for quality control.

Point Estimate

A point estimate gives us a single value as an approximation of a population parameter. It's derived from sample data and helps infer the likely true parameter of the population.

In our example, the engineer calculated the point estimate of the proportion of defective computer chips. Sourcing from the sample of 212 chips, 27 were found to have defects, leading to a point estimate \( \hat{p} = \frac{27}{212} \approx 0.127 \).

This estimate suggests that approximately 12.7% of the total chips produced may be defective. However, remember this is just an estimate. Its accuracy depends on how well the sample represents the entire population.

Standard Error

The standard error is a statistical measure that tells us how much the sample point estimate is expected to differ from the true population parameter. It's critical for understanding the accuracy and reliability of our point estimate.

For proportions, the standard error is calculated using the formula:
\[ SE = \sqrt{ \frac{\hat{p}(1 - \hat{p})}{n} } \]
where \( \hat{p} \) is the sample proportion, and \( n \) is the sample size.

From our step-by-step solution, using \( \hat{p} = 0.127 \) and \( n = 212 \), the standard error was found to be approximately 0.0242. This insight shows the potential variability of our point estimate, offering a window into the certainty of our results.

Thus, standard error allows us to gauge the precision of our estimates and make informed decisions in quality control.

Proportion

A proportion is a fundamental concept in statistics, referring to a part or fraction of the whole. In quality control, knowing the proportion of defects is integral to maintaining high product standards.

The proportion in this scenario, specifically \( \hat{p} = 0.127 \), represents the ratio of defective chips to the total number of chips sampled. This ratio provides insight into the quality of the batch tested, helping the engineer to monitor any deviations from expected performance levels.

Overall, proportions help in benchmarking, setting standards, and evaluating process efficiencies and outputs against historical data or expected norms.

Quality Control

Quality control is an essential practice in manufacturing, aiming to ensure products meet certain standards. It involves processes to monitor, manage, and improve the quality of production.

In the chip production scenario, the engineer uses statistical inference to determine if the defect rate stays within acceptable limits. By comparing the point estimate of defects (12.7%) to the historical rate of 10%, she can assess the process's consistency.

Key steps in quality control include:

• Sampling: Randomly selecting products to inspect for defects.
• Estimation: Using statistics to make inferences about the population quality.
• Decision-making: Implementing changes if the defect rate deviates significantly from expected quality standards.

Thus, quality control ensures ongoing improvements, minimizing defects, and maintaining customer satisfaction.