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 consists of all the computer chips produced in the factory during this week.

02

Identify the Parameter

The parameter being estimated is the true proportion of defective chips in the entire population, denoted as \( p \).

03

Calculate the Point Estimate

The point estimate for the proportion of defective chips is the sample proportion \( \hat{p} \). This is calculated by dividing the number of defective chips by the total number of sampled chips: \( \hat{p} = \frac{27}{212} \approx 0.1274 \).

04

Name of the Statistic

The statistic that measures the uncertainty of the point estimate is the standard error of the sample proportion.

05

Calculate the Standard Error

The standard error (SE) of the sample proportion is calculated using the formula \( SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \), where \( n \) is the sample size. Substitute \( \hat{p} = 0.1274 \) and \( n = 212 \) into the formula:\[SE = \sqrt{\frac{0.1274 \times (1 - 0.1274)}{212}} \approx 0.022 \].

06

Evaluate Engineer's Surprise

Given the historical rate of \(10\%\), compare it with the observed rate of \(12.74\%\). The engineer would not be surprised if the observed proportion's 95\% confidence interval includes \(10\%\). Since the standard error is \(0.022\), the interval is approximately \(12.74\% \pm 4.4\%\), covering the historical rate. Thus, the engineer should not be surprised.

07

Recompute Standard Error with Historical Value

Recompute the standard error using \( p = 0.1 \):\[SE = \sqrt{\frac{0.1 \times (1 - 0.1)}{212}} \approx 0.021 \].The resulting value doesn't change much from the previous \( SE = 0.022 \).

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Key Concepts

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

Population Parameter

The term "population parameter" refers to a specific number that characterizes some aspect of the entire population. In the context of this chip production quality control exercise, the population parameter is the true proportion of defective chips produced in the factory during the week in question. This is a fixed value that provides a comprehensive overview of the defect rate in all chips manufactured during that period. Understanding the population parameter is important because it allows us to make informed decisions about the quality of the product on a larger scale. This value is often unknown, which is why we rely on samples to estimate it.

Point Estimate

A point estimate is a single value that serves as a best guess or estimate of an unknown population parameter based on sample data. In our quality control exercise for computer chips, the point estimate refers to the proportion of defective chips found in the random sample of 212 chips, which is calculated to be approximately 0.1274. This means about 12.74% of the sampled chips were defective. Point estimates provide a concise summary of data and are crucial in making predictions or decisions related to the population. However, they do not convey the uncertainty inherent in sampling, which is why they are often supplemented with additional statistical measures, like confidence intervals, to provide more context.

Standard Error

The standard error quantifies the uncertainty or variability of a point estimate, indicating how much a sample statistic such as the sample proportion can be expected to vary from the true population parameter. For this exercise, the standard error of the sample proportion 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. This value estimates how much the sample proportion might differ from one sample to another, helping us assess the reliability of our point estimate. With a calculated standard error of approximately 0.022, this value helps indicate the range in which the true population parameter (defect rate) may lie.

Sample Proportion

The sample proportion, denoted \( \hat{p} \), is the fraction of individuals in a sample that have a particular attribute of interest. In the computer chips example, the sample proportion \( \hat{p} \) is derived by dividing the number of defective chips by the total number of chips sampled (27 out of 212). This calculation gives us a sample proportion of 0.1274. Understanding the sample proportion is crucial because it acts as a point estimate of the population parameter—the true defect rate in this context. However, the sample proportion, although informative, can vary from sample to sample due to random sampling variability, which is why we compute measures like the standard error to better understand and manage that variability.