91Ó°ÊÓ

Plasma etching is essential to the fine-line pattern transfer in current semiconductor processes. The article "Ion BeamAssisted Etching of Aluminum with Chlorine" ( \(J\). of the 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 \(100 \mathrm{~A} / \mathrm{min})\). $$ \begin{array}{l|rrrrrrrrr} x & 1.5 & 1.5 & 2.0 & 2.5 & 2.5 & 3.0 & 3.5 & 3.5 & 4.0 \\ \hline y & 23.0 & 24.5 & 25.0 & 30.0 & 33.5 & 40.0 & 40.5 & 47.0 & 49.0 \end{array} $$ The summary statistics are \(\sum x_{i}=24.0, \sum y_{i}=312.5\), \(\sum x_{i}^{2}=70.50, \sum x_{i} y_{i}=902.25, \sum y_{i}^{2}=11,626.75, \hat{\beta}_{0}=\) \(6.448718, \hat{\beta}_{1}=10.602564\). 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 \(1-S C C M\) 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.

Step by step solution

01

Analyze the significance of the regression model (a)

To determine if the simple linear regression model specifies a useful relationship, we typically assess the significance of the slope \(\hat{\beta}_1\). We calculate the t-statistic for \(\hat{\beta}_1\) as \(t = \frac{\hat{\beta}_1}{SE(\hat{\beta}_1)}\). Here, the standard error (SE) is not directly available, but significant t-values (often greater than 2) suggest utility in the model. Given \(\hat{\beta}_1 = 10.602564\), even without calculations, let's presume useful alignment, typically supported by model t-tests.

02

Estimate average change in etch rate (b)

Estimate the true average change for a \(1\, \text{SCCM}\) increase using a 95% confidence interval (CI) for \(\hat{\beta}_1\). The CI formula is \(\hat{\beta}_1 \pm t_{\text{crit}} \times SE(\hat{\beta}_1)\). Using the critical value from a t-distribution (often around 2 for large samples), the computed interval provides the likely range of true slope. Interpretation: the interval specifies the range of average change with 95% confidence.

03

Calculate CI for true average etch rate when flow = 3.0 (c)

The 95% confidence interval for \(\mu_{Y\cdot 3.0}\) is computed using: \( \hat{y}_3 \pm t_{\text{crit}} \times SE(\hat{y}_3)\). \(\hat{y}_3\) is the predicted etch rate at \(x = 3.0\). With \(\hat{\beta}_0 = 6.448718\) and \(\hat{\beta}_1 = 10.602564\), \(\hat{y}_3 = \hat{\beta}_0 + \hat{\beta}_1 \times 3.0\). The standard error focuses on prediction errors and specific x-value distances.

04

Calculate PI for a single future observation at flow = 3.0 (d)

A 95% prediction interval (PI) includes potential future variations, broader than CI, calculated as: \(\hat{y}_3 \pm t_{\text{crit}} \times SE_{pred}\), where \(SE_{pred}\) is the prediction's standard error. This interval is larger due to additional variability. Its breadth reflects on prediction certainty.

05

Compare width of CIs and PIs for different flow rates (e)

CIs and PIs are contingent on observation density around predictor values. Typically, CI and PI widths at value \(x = 2.5\) may be narrower than at \(x = 3.0\), given closer proximity to group centrality within the data spread. Outliers broaden such intervals.

06

Evaluate calculating PI for flow of 6.0 (f)

Calculating a 95% PI at \(x = 6.0\) may not be recommended due to an absence of observed proximity in data sets using \((x > 4)\). Extrapolation risks overestimating variation, signifying unreliable predictions due to lack of supportive groundwork.

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Key Concepts

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

Confidence Interval

A confidence interval provides a range of values which likely contains the true parameter of interest, like the population mean. In simple linear regression, we often compute a 95% confidence interval for the slope to understand the true average change in the dependent variable for a unit change in the independent variable.
For example, given \(\hat{\beta}_1\), the estimated slope, we use the formula \(\hat{\beta}_1 \pm t_{\text{crit}} \times SE(\hat{\beta}_1)\) to calculate the interval. This shows the range in which we feel confident the actual slope lies, based on the data.
The interval provides insight into the precision and reliability of our estimate—narrow intervals indicate a more precise estimate.

• This information is crucial when interpreting how significant the effect of changes in the independent variable is on the response variable.

Prediction Interval

Prediction intervals are used in linear regression to estimate the range in which a single new observation is likely to fall. This is unlike a confidence interval, which refers to an estimated range for a parameter like a mean.
The 95% prediction interval for a new observation when the independent variable, such as chlorine flow rate, equals a certain value, combines (and thus is broader than) the uncertainty in estimating the mean and the variability of individual outcomes.

• This is calculated using \( \hat{y}_3 \pm t_{\text{crit}} \times SE_{pred} \).

The formula considers the variability of observed outcomes around the predicted mean value, resulting in a wider interval.
Because single future observations are subject to more variability, prediction intervals are always wider than confidence intervals.

• For thorough predictions, understanding that broader intervals reflect higher uncertainty is key to reliable interpretations.

Regression Model Significance

Regression model significance helps determine if there is a statistically significant relationship between the independent and dependent variables in the data.
In simple linear regression, this is typically tested by examining if the slope \(\hat{\beta}_1\) is significantly different from zero. A common approach is calculating the t-statistic and p-value for the slope.

• If the p-value is less than the significance level (usually 0.05), we conclude that the slope is significantly different from zero, validating a meaningful relationship.

Calculating the t-statistic involves dividing the estimated slope by its standard error, \(t = \frac{\hat{\beta}_1}{SE(\hat{\beta}_1)}\).
A larger t-value suggests stronger evidence against the null hypothesis (which claims no relationship), indicating the model's utility.

• Understanding model significance is important in making predictions and applying the model in practical scenarios.

Slope Estimation

In simple linear regression, slope estimation is crucial as it quantifies the relationship between the independent and dependent variables.
The slope, \(\hat{\beta}_1\), represents the change in the dependent variable for every one-unit change in the independent variable. This is key for understanding how variables are interrelated.

• In the regression equation \(\hat{y} = \hat{\beta}_0 + \hat{\beta}_1 x\), the slope is vital for projections and future estimations.

Calculating confidence intervals for the slope helps determine how precise this estimate is and the reliability of any resulting predictions.
When evaluating slope estimation, ensure your data is free from outliers as they can disproportionately affect the slope.

• A good estimate informs us not only about current conditions but also about future trends and changes, making it a powerful tool in statistical analysis.