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 \(\mu_{\gamma \cdot 75}\), 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 \(\Sigma y_{i}^{2}=8386.43, \Sigma y_{i}=210.70, \Sigma x_{i} y_{i}=17,002.00\), and \(\Sigma x_{i}^{2} y_{i}=1,419,780\), compute SSE \(\left[=\Sigma y_{i}^{2}-\right.\) \(\left.\hat{\beta}_{0} \Sigma y_{i}-\hat{\beta}_{1} \Sigma x_{i} y_{i}-\hat{\beta}_{2} \Sigma x_{i}^{2} y_{i}\right]\) 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 \(s_{\hat{\beta}_{2}}=.00226\), test \(H_{0}: \beta_{2}=0\) versus \(H_{2}: \beta_{2} \neq 0\) at level \(.01\), and interpret the result.

Step by step solution

01

Estimate \\mu_{\\gamma \\cdot 75} for \\gamma=75 \\mathrm{rpm}

To estimate \( \mu_{\gamma \cdot 75} \), substitute \( x = 75 \) into the quadratic regression equation: \[ y = -113.0937 + 3.3684 \times 75 - 0.01780 \times 75^2 \]. This computes to \( y = 113.093 \).

02

Predict viscosity for speed 60 \\mathrm{rpm}

Use the quadratic regression equation with \( x = 60 \): \[ y = -113.0937 + 3.3684 \times 60 - 0.01780 \times 60^2 \]. Solve to predict \( y = -113.0937 + 202.104 - 64.08 = 24.9303 \).

03

Compute SSE

Find SSE using: \[ \text{SSE} = \Sigma y_{i}^{2} - \hat{\beta}_{0} \Sigma y_{i} - \hat{\beta}_{1} \Sigma x_{i} y_{i} - \hat{\beta}_{2} \Sigma x_{i}^{2} y_{i} \]. Plugging in the values: \[ \text{SSE} = 8386.43 - (-113.0937 \times 210.70) - (3.3684 \times 17002.00) - (-0.01780 \times 1419780) \]. Compute the values to find \( \text{SSE} = 49.088 \).

04

Compute standard error s

The standard error \( s \) is given by \( s = \sqrt{\frac{\text{SSE}}{n-3}} \). With \( \text{SSE} = 49.088 \) and \( n = 6 \), compute \( s = \sqrt{\frac{49.088}{3}} \). This results in \( s = 4.05 \).

05

Calculate R-squared value

The R-squared formula is \( R^2 = 1 - \frac{\text{SSE}}{\text{SST}} \). With \( \text{SSE} = 49.088 \) and \( \text{SST} = 987.35 \), calculate \( R^2 = 1 - \frac{49.088}{987.35} = 0.9503 \).

06

Test hypothesis for \\beta_{2}

Compute \( t \)-statistic for \( \beta_{2} \): \( t = \frac{\hat{\beta}_{2}}{s_{\hat{\beta}_{2}}} \) with \( \hat{\beta}_{2} = -0.01780 \) and \( s_{\hat{\beta}_{2}} = 0.00226 \). Calculate \( t = \frac{-0.01780}{0.00226} = -7.876 \). Compare with critical \( t \) value from \( t \)-table for \( df = 3 \) at \( 0.01 \) significance to find \( |t| \) exceeds critical value, indicating rejection of \( H_0 \).

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.

Simple Linear Regression

Simple linear regression is a statistical method used to examine the relationship between two continuous variables. In its simplest form, it models the dependent variable (often denoted as \(y\)) as a function of an independent variable (\(x\)), typically taking the form \( y = \beta_0 + \beta_1x \), where \( \beta_0 \) is the y-intercept and \( \beta_1 \) is the slope.

The goal is to find the slope \( \beta_1 \) and the intercept \( \beta_0 \) that minimize the sum of the square differences between the observed values and those predicted by the line. This method is different from quadratic regression, which includes higher powers of the independent variable, allowing for curvature in the relationship rather than a straight line.

While simple linear regression can be very effective for linear relationships, if data show a curvilinear trend, like in the quadratic regression case described in the original exercise, simple linear regression may not be suitable. Instead, we use quadratic regression to accommodate the curvature.

Standard Error

The standard error (SE) is a measure of the accuracy of predictions made by a regression model. It indicates how much a set of observations varies around the regression line. Specifically, it helps us to estimate the average distance of the observed values from the estimated value predicted by the regression line.

In the context of the given exercise, we calculated the standard error of the estimate using the formula \( s = \sqrt{\frac{\text{SSE}}{n-3}} \), where SSE is the sum of squared errors, and \( n \) is the number of data points. In this case, we found that \( s = 4.05 \).

A smaller standard error suggests that the model's predictions are closer to the actual data points. Conversely, a larger standard error means there is more variability and less reliability in the model's predictions.

T-test

A t-test is a type of statistical test used to compare the means of two groups and can also be used to determine if a specific coefficient in a regression model is statistically significant. In the context of this exercise, we use a t-test to evaluate the hypothesis about the coefficient \( \beta_{2} \).

The calculation involves determining the \( t \)-statistic through the formula \( t = \frac{\hat{\beta}_{2}}{s_{\hat{\beta}_{2}}} \), where \( \hat{\beta}_{2} \) is the estimated coefficient, and \( s_{\hat{\beta}_{2}} \) is the standard error of that coefficient. We obtained \( t = -7.876 \) for our case.

By comparing the calculated \( t \)-statistic to a critical value from the t-distribution table, we can decide whether or not to reject the null hypothesis. In this exercise, the high value of the t-statistic indicates that \( \beta_{2} \) is significantly different from zero.

Hypothesis Testing

Hypothesis testing is a statistical method that allows us to make decisions about a population based on sample data. The process involves formulating two statements, the null hypothesis (\( H_0 \)) and the alternative hypothesis (\( H_a \)), and using statistical evidence to make inferences about which hypothesis is more plausible.

In the quadratic regression context of this exercise, we tested the null hypothesis \( H_0: \beta_{2} = 0 \) against the alternative hypothesis \( H_a: \beta_{2} eq 0 \). Essentially, this test asks whether the quadratic term contributes significantly to the model.

After computing the t-statistic and referencing critical values in the t-distribution table at the 0.01 significance level, we found that the calculated \( t \)-statistic exceeded the critical value, leading to rejecting the null hypothesis. This result suggests that there is sufficient evidence to conclude that the quadratic term is indeed a significant part of the model.