91Ó°ÊÓ

Plasma etching is essential to the fine-line pattern transfer in current semiconductor processes. The article "Ion Beam-Assisted Etching of Aluminum with Chlorine" (J. Electrochem. Soc., 1985: 2010-2012) gives the accompanying data (read from a graph) on chlorine flow \((x\), in SCCM) through a nozzle used in the etching mechanism and etch rate \((y\), in \(100 \mathrm{~A} / \mathrm{min})\). $$ \begin{array}{l|lrrrrrrrr} x & 1.5 & 1.5 & 2.0 & 2.5 & 2.5 & 3.0 & 3.5 & 3.5 & 4.0 \\ \hline y & 23.0 & 24.5 & 25.0 & 30.0 & 33.5 & 40.0 & 40.5 & 47.0 & 49.0 \end{array} $$ a. Does the simple linear regression model specify a useful relationship between chlorine flow and etch rate? b. Estimate the true average change in etch rate associated with a 1-SCCM increase in flow rate using a \(95 \%\) confidence interval, and interpret the interval. c. Calculate a \(95 \%\) CI for \(\mu_{Y \cdot 3.0}\), the true average etch rate when flow \(=3.0\). Has this average been precisely estimated? d. Calculate a \(95 \%\) PI for a single future observation on etch rate to be made when flow \(=3.0 .\) Is the prediction likely to be accurate? e. Would the \(95 \%\) CI and PI when flow \(=2.5\) be wider or narrower than the corresponding intervals of parts (c) and (d)? Answer without actually computing the intervals. f. Would you recommend calculating a \(95 \%\) PI for a flow of 6.0? Explain. g. Calculate simultaneous CI's for true average etch rate when chlorine flow is \(2.0,2.5\), and \(3.0\), respectively. Your simultaneous confidence level should be at least \(97 \%\).

Step by step solution

01

Compute the Simple Linear Regression Model

The linear regression model is given by \( y = \beta_0 + \beta_1 x + \epsilon \). To fit this model to the data, we calculate the slope \( \beta_1 \) and intercept \( \beta_0 \) using least squares estimation. Using the given data, calculate the correlation coefficient and the equation of the regression line.

02

Assessing Model Usefulness (Part a)

Assess the usefulness of the regression model by calculating the coefficient of determination \( R^2 \). \( R^2 \) provides the proportion of variance in the dependent variable that is predictable from the independent variable. A higher \( R^2 \) value indicates a stronger relationship.

03

Estimate the Change in Etch Rate (Part b)

Calculate the 95% confidence interval for \( \beta_1 \), which is the slope of the regression line. This interval gives us an estimate for the change in etch rate for each additional SCCM of chlorine flow. Use the standard error of the slope and the critical t-value to compute this interval.

04

Compute the Confidence Interval for \( \mu_{Y \cdot 3.0} \) (Part c)

Use the regression output to find the estimate of the mean response when \( x = 3.0 \). Compute the 95% confidence interval for this mean response using the formula for the confidence interval of a mean predicted value. Evaluate the precision based on the width of the interval.

05

Compute the Prediction Interval for a Future Observation (Part d)

Calculate the 95% prediction interval for a single future observation when \( x = 3.0 \). Use the prediction interval formula, which accounts for both the error in estimating the population mean and the variability of the new observation. A wider interval indicates less accuracy.

06

Compare Intervals for \( x = 2.5 \) to \( x = 3.0 \) (Part e)

Without calculation, determine if the intervals when \( x = 2.5 \) would be wider or narrower compared to those at \( x = 3.0 \). Since \( x = 2.5 \) is closer to the mean of \( x \) values, intervals at \( x = 2.5 \) are expected to be narrower because the prediction error decreases near the mean.

07

Evaluate Sufficiency for PI at Flow 6.0 (Part f)

Consider the recommendation for calculating a 95% PI at a flow rate of 6.0. Since 6.0 is outside the range of provided data, extrapolation increases uncertainty, making the prediction interval potentially unreliable. Hence, it's generally not advisable to calculate such a PI.

08

Calculate Simultaneous Confidence Intervals (Part g)

Use the regression equation to compute estimates for \( \mu_y \) at \( x = 2.0, 2.5, \text{ and } 3.0 \). Apply a method like Bonferroni's correction to obtain simultaneous confidence intervals that account for multiple comparisons, ensuring a confidence level of at least 97%.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

Confidence Interval

Confidence intervals are crucial in regression analysis because they provide a range in which we expect the true parameter, like a mean or a slope, to lie with a certain level of confidence. For instance, when estimating the mean response (etch rate) for a specific chlorine flow, say 3.0 SCCM, a 95% confidence interval gives a range where the true mean etch rate is likely to be found.
This interval is calculated using the regression output including the estimated mean and the standard error, combined with a critical value from the t-distribution. The narrower the interval, the more precise our estimate is. However, exact precision depends on factors like the sample size and the variability of the data.
When comparing confidence intervals for different flow rates, the width changes depending on how far the point is from the mean of the sample. Points closer to the sample mean generally have narrower intervals, indicating less estimation error.

Prediction Interval

A prediction interval is similar to a confidence interval, but it’s used to predict the interval within which a single future observation will fall. In our context, if we want to predict a future etch rate at 3.0 SCCM, a 95% prediction interval is calculated.
It accounts for the variability not just of the mean response, like a confidence interval, but also the data scatter around the regression line. This makes prediction intervals wider than confidence intervals. Key to its calculation are the standard error of the estimate and an appropriate t-value.
If predictions are made far from the central range of the initially observed data, prediction intervals become less reliable. For example, extending the prediction to a chlorine flow of 6.0 SCCM would involve a high degree of untrustworthiness due to extrapolation beyond the existing data range.

Simple Linear Regression

In simple linear regression, we model the relationship between two variables by fitting a linear equation to the data. This relationship is represented by the formula: \( y = \beta_0 + \beta_1 x + \epsilon \) where \( y \) is the response variable, \( \beta_0 \) is the intercept, \( \beta_1 \) is the slope, \( x \) is the predictor variable, and \( \epsilon \) is the error term. The main task is to estimate the parameters \( \beta_0 \) and \( \beta_1 \) using methods like least squares estimation, which minimizes the sum of the squared differences between observed and predicted values. Once we have these estimates, we can make inferences about the relationship between chlorine flow and etch rate.
These inferences include evaluating the strength (with \( R^2 \)) and significance (with t-tests) of the relationship, estimating changes in etch rate per unit increase in flow, and making predictions at given flow rates.

Coefficient of Determination

The coefficient of determination, denoted as \( R^2 \), is a vital statistic in regression analysis. It tells us what proportion of the total variability in the response variable is explained by the predictor variable. An \( R^2 \) of 1 indicates perfect correlation, whereas an \( R^2 \) of 0 suggests no linear relationship.
Calculating \( R^2 \) involves dividing the sum of squares due to regression by the total sum of squares. A higher \( R^2 \) value, typically close to 1, reflects a strong relationship between the chlorine flow and etch rate in our study. This metric helps in assessing model utility; a high \( R^2 \) means our regression model reliably explains the variation in etch rates.
However, it's important to remember that a high \( R^2 \) doesn't imply causation. It's merely indicative of correlation within the given data range and conditions.