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The article "Analysis of the Modeling Methodologies for Predicting the Strength of Air-Jet Spun Yarns" (Textile Res. \(J ., 1997: 39-44\) ) reported on a study carried out to relate yarn tenacity \((y\), in \(\mathrm{g} /\) tex) to yarn count ( \(x_{1}\), in tex), percentage polyester \(\left(x_{2}\right)\), first nozzle pressure \(\left(x_{3}\right.\), in \(\left.\mathrm{kg} / \mathrm{cm}^{2}\right)\), and second nozzle pressure \(\left(x_{4}\right.\), in \(\left.\mathrm{kg} / \mathrm{cm}^{2}\right) .\) The estimate of the constant term in the corresponding multiple regression equation was \(6.121\). The estimated coefficients for the four predictors were \(-.082, .113, .256\), and \(-.219\), respectively, and the coefficient of multiple determination was .946. Assume that \(n=25\). a. State and test the appropriate hypotheses to decide whether the fitted model specifies a useful linear relationship between the dependent variable and at least one of the four model predictors. b. Calculate the value of adjusted \(R^{2}\) and comment. c. Calculate a \(99 \%\) confidence interval for true mean yarn tenacity when yarn count is 16.5, yarn contains \(50 \%\) polyester, first nozzle pressure is 3 , and second nozzle pressure is 5 if the estimated standard deviation of predicted tenacity under these circumstances is \(.350\).

Step by step solution

01

Formulating Hypotheses for Part (a)

The null hypothesis for the F-test is: \( H_0: \beta_1 = \beta_2 = \beta_3 = \beta_4 = 0 \), implying no linear relationship. The alternative hypothesis is \( H_a: \) at least one \( \beta_i eq 0 \), indicating a useful model.

02

Conducting F-test for Part (a)

To conduct the F-test, the test statistic is computed as \[ F = \frac{R^2 / (k)}{(1-R^2) / (n-k-1)} \] where \( R^2 = 0.946 \), \( n = 25 \), and \( k = 4 \). Substitute these values into the formula: \[ F = \frac{0.946 / 4}{(1-0.946) / (25-5)} = \frac{0.2365}{0.0027} \approx 87.59 \]. Compare this with the critical value from the F-table with 4 and 20 degrees of freedom at a specific significance level.

03

Deciding the Outcome of the F-test

Since the calculated F-statistic \(87.59\) is typically greater than the critical value (for a significance level of 0.05), we reject the null hypothesis. Thus, the model has a useful linear relationship with the dependent variable.

04

Calculating Adjusted R-Squared for Part (b)

Adjusted \( R^2 \) accounts for the number of predictors and is calculated by the formula: \[ R^2_{adj} = 1 - \frac{(1 - R^2)(n - 1)}{n - k - 1} \] Substituting \( R^2 = 0.946, n = 25, k = 4 \): \[ R^2_{adj} = 1 - \frac{(1 - 0.946)(24)}{20} = 1 - \frac{1.296}{20} = 0.937 \]. This indicates that about 93.7% of the variability is explained by the model.

05

Confidence Interval Calculation for Part (c)

The 99% confidence interval for the mean yarn tenacity is: \[ y_0 \pm t* \times s = (6.121 - 0.082\times16.5 + 0.113\times50 + 0.256\times3 - 0.219\times5) \pm t* \times 0.350 \]. Compute the point estimate: \( y_0 = 6.121 - 1.353 + 5.65 + 0.768 - 1.095 = 10.091 \). The critical value \( t* \) for 20 degrees of freedom at a 99% confidence level is approximately 2.845. Compute the interval: \[ 10.091 \pm 2.845 \times 0.350 = [9.099, 11.083] \].

06

Interpreting Confidence Interval Result

The 99% confidence interval for the mean yarn tenacity when the given predictors are used is \([9.099, 11.083]\). This suggests that we can be 99% confident that the true mean yarn tenacity for the specified conditions falls within this interval.

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Key Concepts

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

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about a population based on sample data. In the context of multiple regression analysis, hypothesis testing involves assessing whether the relationship between the dependent and independent variables is statistically significant.

To decide if a fitted model signifies a valid linear relationship, we often employ the F-test. In this case, the null hypothesis, denoted as \(H_0\), proposes that all regression coefficients \(\beta_1, \beta_2, \beta_3, \) and \(\beta_4\) are zero. This implies no linear relationship between the independent and dependent variables. Conversely, the alternative hypothesis \(H_a\) suggests that at least one coefficient is non-zero, indicating a meaningful model.

For example, in a multiple regression scenario with an \(R^2\) value of 0.946, the F-test statistic can be calculated using:

• \(k\), the number of independent variables (4 in this case)
• \(n\), the number of observations (25 here)
• The formula: \[F = \frac{R^2 / k}{(1-R^2) / (n-k-1)}\]

An F-statistic much larger than the critical value for a given significance level means rejecting the null hypothesis, confirming the model's validity.

Coefficient of Determination

The Coefficient of Determination, often represented as \(R^2\), is a crucial metric in regression analysis. It indicates the proportion of the variation in the dependent variable that can be explained by the independent variables in the model.

Simply put, \(R^2\) measures the model's goodness of fit. A value close to 1 suggests that a large proportion of the variability in the response variable is accounted for by the predictors. In the provided exercise, an \(R^2\) of 0.946 implies that around 94.6% of the variability in yarn tenacity can be explained by yarn count, percentage polyester, and nozzle pressures.

However, it's important to acknowledge that \(R^2\) alone doesn't provide information about the model's accuracy or whether it includes superfluous variables. This is why other metrics, such as the adjusted \(R^2\), are also considered, especially when multiple predictors are involved.

Adjusted R-squared

Adjusted \(R^2\) offers an improved version of the Coefficient of Determination by adjusting for the number of predictors in the model. It effectively penalizes the addition of variables that don't increase the explanatory power substantially.

This metric is especially informative in models with multiple independent variables, as it can help identify if new predictors are truly contributing meaningful information or just inflating the \(R^2\) value due to additional parameters.

The calculation for adjusted \(R^2\) is:\[R^2_{adj} = 1 - \frac{(1 - R^2)(n - 1)}{n - k - 1}\]where:

• \(n\) is the number of observations,
• \(k\) is the number of predictors.

In the given exercise, the adjusted \(R^2\) was approximately 0.937, indicating that about 93.7% of the variability is explained by the predictors, considering the complexity of the model.

Adjusted \(R^2\) provides a clearer picture of a model's efficiency when dealing with multiple variables.

Confidence Intervals

Confidence intervals provide a range of values within which we can be certain the true parameter value falls. They are essential in understanding the precision and reliability of an estimate in statistical modeling.

In regression analysis, a confidence interval can be used to estimate the true mean of the dependent variable for specific values of predictors. For instance, in our exercise, a 99% confidence interval was calculated for yarn tenacity under specified conditions of yarn count, polyester percentage, and nozzle pressures.

The calculation of a 99% confidence interval involves:

• Point estimate of the dependent variable
• Standard deviation of the estimate (\(s = 0.350\) in the example)
• Critical t-value (based on the confidence level and degrees of freedom)

Using the formula: \[y_0 \pm t* \times s\]This interval provides a high degree of certainty (99%) about the true mean yarn tenacity, computed to be \([9.099, 11.083]\) in the specified scenario.

Confidence intervals are powerful as they convey statistical confidence about estimated ranges, helping to make more informed decisions based on data.