91Ó°ÊÓ

The viscosity \((y)\) of an oil was measured by a cone and plate viscometer at six different cone speeds \((x)\). It was assumed that a quadratic regression model was appropriate, and the estimated regression function resulting from the \(n=6\) observations was $$ y=-113.0937+3.3684 x-.01780 x^{2} $$ a. Estimate \(\mu_{\gamma .75}\), the expected viscosity when speed is \(75 \mathrm{rpm} .\) b. What viscosity would you predict for a cone speed of \(60 \mathrm{rpm}\) ? c. If \(\sum y_{i}^{2}=8386.43, \Sigma y_{j}=210.70, \Sigma x_{i} y_{i}=17,002.00\), and \(\sum x_{1}^{2} y_{i}=1,419,780\), compute \(\mathrm{SSE}\left[=\sum y_{i}^{2}-\right.\) \(\left.\hat{\beta}_{0} \Sigma y_{i}-\hat{\beta}_{1} \Sigma x_{i} y_{s}-\hat{\beta}_{2} \Sigma x_{i}^{2} y_{i}\right]\) and \(s\). d. From part (c), SST \(=8386.43-(210.70)^{2} / 6=987.35\). Using SSE computed in part (c), what is the computed value of \(R^{2} ?\) e. If the estimated standard deviation of \(\hat{\beta}_{2}\) is \(s_{\dot{\beta}}=.00226\), test \(H_{0}: \beta_{2}=0\) versus \(H_{\mathrm{a}}: \beta_{2} \neq 0\) at level 01 , and interpret the result.

Step by step solution

01

Estimate the Expected Viscosity for 75 RPM

To estimate \( \mu_{\gamma .75} \), substitute \( x = 75 \) into the regression function: \( y = -113.0937 + 3.3684 \times 75 - 0.01780 \times 75^2 \). Calculate \( y = -113.0937 + 253.63 - 100.0125 = 40.5238 \).So, the expected viscosity when the speed is 75 RPM is approximately 40.52.

02

Predict the Viscosity for 60 RPM

Substitute \( x = 60 \) into the regression function to predict viscosity: \[ y = -113.0937 + 3.3684 \times 60 - 0.01780 \times 60^2 \]. \( y = -113.0937 + 202.104 - 64.08 = 24.9303 \).Thus, the predicted viscosity at 60 RPM is approximately 24.93.

03

Compute SSE (Sum of Squares for Error)

Use the formula for SSE: \( \text{SSE} = \sum y_{i}^{2} - \hat{\beta}_{0} \Sigma y_{i} - \hat{\beta}_{1} \Sigma x_{i} y_{i} - \hat{\beta}_{2} \Sigma x_{i}^{2} y_{i} \).Plug in the given values: \[ \text{SSE} = 8386.43 - (-113.0937 \times 210.70) - (3.3684 \times 17002) - (-0.01780 \times 1419780) \]. Calculate to get SSE = 19.38.

04

Calculate s (Standard Error of the Estimate)

The standard error \( s \) is calculated as \( s = \sqrt{\frac{\text{SSE}}{n-p}} \), where \( n = 6 \) (the number of observations) and \( p = 3 \) (the number of parameters in the model). Thus, \( s = \sqrt{\frac{19.38}{6-3}} = \sqrt{6.46} = 2.54 \).The value of \( s \) is approximately 2.54.

05

Compute R² (Coefficient of Determination)

To compute \( R^2 \), use the formula \( R^2 = 1 - \frac{\text{SSE}}{\text{SST}} \).Given \( \text{SST} = 987.35 \), and \( \text{SSE} = 19.38 \), \( R^2 = 1 - \frac{19.38}{987.35} \).Calculate \( R^2 = 1 - 0.01963 = 0.98037 \).So, \( R^2 \approx 0.98 \), meaning the model explains 98% of the variance.

06

Test the Hypothesis for β2

To test \( H_0: \beta_2 = 0 \) versus \( H_a: \beta_2 eq 0 \), calculate the test statistic: \( t = \frac{\hat{\beta}_{2} - 0}{s_{\hat{\beta}_{2}}} = \frac{-0.01780}{0.00226} = -7.876 \).With degrees of freedom \( n-p = 3 \), check the t-distribution at 0.01 level for significance.Since \( |t| \) is greater than the critical value, we reject \( H_0 \). The result suggests that \( \beta_2 \) is significantly different from 0.

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Key Concepts

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

Expected Viscosity Calculation

To determine the expected viscosity at a specific speed, you need to substitute the given speed value into the regression model equation. For example, if you're asked to find the expected viscosity at 75 RPM, you would plug 75 in place of \( x \) in the equation: \( y = -113.0937 + 3.3684 \times 75 - 0.01780 \times 75^2 \).
This will yield a result which, in this case, is approximately 40.52. This value represents the estimated viscosity the oil would have at 75 RPM. It is important when performing these calculations to follow the order of operations (PEMDAS/BODMAS) to ensure accuracy.
This method helps predict and understand how viscosity changes with varying speeds, an essential part of Quadratic Regression Analysis.

SSE (Sum of Squares for Error)

Sum of Squares for Error (SSE) is a crucial component in determining how well a regression model fits the data. It is calculated by subtracting the sum of the squares of the differences between each observed value and the value predicted by the model.
SSE essentially measures the variance in the data that cannot be explained by the model. In our example, the formula for SSE is given by \( \text{SSE} = \sum y_{i}^{2} - \hat{\beta}_{0} \Sigma y_{i} - \hat{\beta}_{1} \Sigma x_{i} y_{i} - \hat{\beta}_{2} \Sigma x_{i}^{2} y_{i} \). By substituting known values into this formula, we get an SSE of 19.38.
A smaller SSE value indicates a better fit of the regression model. It shows that the differences between the observed and predicted viscosities are minimal.

Coefficient of Determination (R²)

The Coefficient of Determination, represented as \( R^2 \), is a key metric used to assess the explanatory power of a regression model. The \( R^2 \) value indicates the proportion of the variance in the dependent variable that is predictable from the independent variable(s).
In our case, the \( R^2 \) is calculated using \( R^2 = 1 - \frac{\text{SSE}}{\text{SST}} \). Given values of \( \text{SSE} = 19.38 \) and \( \text{SST} = 987.35 \), \( R^2 = 1 - 0.01963 = 0.98037 \).
Thus, an \( R^2 \) of approximately 0.98 implies that the quadratic regression model explains 98% of the variability in the viscosity data. This high value suggests that the model is a very good fit.

Hypothesis Testing for Regression Coefficients

Hypothesis testing in the context of regression coefficients involves testing whether these coefficients are significantly different from zero. By testing \( \beta_2 \), we determine if there is a significant quadratic effect on viscosity.
The process starts with the null hypothesis \( H_0 \): \( \beta_2 = 0 \) and the alternative hypothesis \( H_a \): \( \beta_2 eq 0 \). For this test, the formula to compute the test statistic \( t \) is \( t = \frac{\hat{\beta}_{2} - 0}{s_{\hat{\beta}_{2}}} \), which is calculated as \( t = \frac{-0.01780}{0.00226} = -7.876 \).
This test statistic is then compared against a critical value from the t-distribution at the chosen significance level (0.01 here). Given that \(|t|\) is greater than this critical value, we conclude that there is significant evidence to reject the null hypothesis. Thus, \( \beta_2 \) is statistically significant, indicating a meaningful quadratic relationship in the viscosity-speed data.