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Chapter 13: Problem 3

Suppose that you fit the model $$ E(y)=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{2}+\beta_{3} x_{3} $$ to 15 data points and found \(F\) equal to 57.44 . a. Do the data provide sufficient evidence to indicate that the model contributes information for the prediction of \(y\) ? Test using a \(5 \%\) level of significance. b. Use the value of \(F\) to calculate \(R^{2}\). Interpret its value.

Short Answer

Expert verified
Based on the given information, we conclude that the multiple regression model is significant in contributing information for the prediction of y at a 5% level of significance since the calculated F value (57.44) is greater than the critical value (4.07). The R虏 value, which measures how well the variation in the dependent variable y is explained by the model, is approximately 0.940 or 94%. This means that the model explains about 94% of the variation in the dependent variable, indicating a strong relationship between the independent variables and the dependent variable.

Step by step solution

01

Perform an F-test

First, we need to perform the F-test using the given F value, 57.44, and a 5% level of significance. In this case, we have 3 independent variables and 11 degrees of freedom for the error. We can find the critical value of F using a table or a calculator for the F-distribution. We need to compare our F value, 57.44, with the critical value of F for the given degrees of freedom (3 numerator and 11 denominator) and a 5% level of significance. If our calculated F value is greater than the critical value, we can conclude that the model contributes significant information for the prediction of y. Using a calculator or a table, the critical value for F(3, 11) at 5% level of significance is approximately 4.07. Since 57.44 > 4.07, we can conclude that the data provides sufficient evidence to indicate that the model contributes information for the prediction of y at a 5% level of significance.
02

Calculate R虏

Next, let's calculate R虏. Given the F value, we can use the following formula to find the R虏 value: $$ R^2 = \frac{F \times k}{F \times k + (n - k - 1)} $$ Where: - \(R^2\) is the coefficient of determination - \(F\) is the calculated F value (57.44) - \(k\) is the number of independent variables (3) - \(n\) is the total number of data points (15) Plugging the values into the formula, we get: $$ R^2 = \frac{57.44 \times 3}{57.44 \times 3 + (15 - 3 - 1)} = \frac{172.32}{172.32 + 11} \approx 0.940 $$ So, R虏 is approximately 0.940, or 94%.
03

Interpretation of R虏

The value of R虏 is a measure of how well the variation in the dependent variable y is explained by the independent variables in the model. In this case, with an R虏 of 0.940, we can say that our model explains about 94% of the variation in y. This indicates a very strong relationship between the independent variables and the dependent variable in the model.

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Key Concepts

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

F-test
The F-test is a statistical test used to determine if there are significant relationships between variables in a linear regression model. In simple terms, we want to know if our independent variables collectively have a meaningful impact on the dependent variable.

To conduct an F-test, you compare your calculated F value to a critical F value from F-distribution tables. You need certain information: the number of independent variables (numerator degrees of freedom) and the degrees of freedom for the error (denominator degrees of freedom). In our exercise, these values are 3 and 11, respectively.

The logic is straightforward: if the calculated F is greater than the critical F value, it implies that the regression model provides significant predictions about the dependent variable. Given the F value (57.44) from the exercise and a critical value of 4.07 with a 5% significance level, we conclude that the model is significant. That means it adds value to our predictive ability.
Coefficient of determination
The coefficient of determination, represented as \( R^2 \), is an essential metric in linear regression. It tells us how much variance in the dependent variable is explained by the independent variables in the model.

In essence, if you think of variance as the amount of spread or variability in your data, \( R^2 \) quantifies how much of this variability your model accounts for.

An \( R^2 \) value closer to 1 indicates a strong link; in our case, an \( R^2 \) of 0.940 signifies that the model explains 94% of the variance in the outcome variable, \( y \). A high \( R^2 \) like this suggests your predictors are doing an excellent job capturing the data patterns, leading to robust predictions.
Level of significance
The level of significance, denoted by \( \alpha \), defines the threshold for deciding if a test result is statistically significant. It's about balancing the risk of falsely claiming a difference or effect when none exists (Type I error).

In hypothesis testing, you use the level of significance to determine if your findings support rejecting the null hypothesis. Commonly set at 0.05 (or 5%), it implies you're willing to accept a 5% risk of claiming a predictive relationship between variables when there might be none.

In our F-test scenario, a significance level of 5% juxtaposed with an F value surpassing the critical threshold affirms the model's predictive value. It points out that the likelihood of the observed association being due to mere chance is minimal.
Degrees of freedom
Degrees of freedom (df) are a concept in statistical calculations that reflects the number of independent values in a dataset that can vary while estimating statistical parameters. For instance, in our F-test, degrees of freedom help determine the critical value needed to assess the significance of our test statistic.

Typically, degrees of freedom depend on the sample size and the number of parameters estimated. For the F-test in linear regression:
  • The numerator degrees of freedom = the number of predicted parameters = number of independent variables (3 in our case).
  • The denominator degrees of freedom = total number of observations minus the number of parameters estimated (including the constant term).

Understanding degrees of freedom allows accurate test interpretations and ensures your results are trustworthy, providing context to how much information was effectively used in your analysis.

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Most popular questions from this chapter

In an investigation to determine the relationship between the degree of metal corrosion and the length of time the metal is exposed to the action of soil acids, the percentage of corrosion and exposure time were measured weekly. $$ \begin{array}{l|llllllll} y & 0.1 & 0.3 & 0.5 & 0.8 & 1.2 & 1.8 & 2.5 & 3.4 \\ \hline x & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \end{array} $$ The data were fitted using the quadratic model, $E(y)=\beta_{0}+\beta_{1} x+\beta_{2} x^{2},$ with the following results. $$ \begin{array}{lr} {\text { SUMMARY OUTPUT }} \\ \hline {\text { Regression Statistics }} \\ \hline \text { Multiple R } & 0.9993 \\ \text { R Square } & 0.9985 \\ \text { Adjusted R Square } & 0.9979 \\ \text { Standard Error } & 0.0530 \\ \text { Observations } & 8 \end{array} $$ $$ \begin{aligned} &\text { ANOVA }\\\ &\begin{array}{lrrrrr} \hline & d f & \mathrm{SS} & M S & F & \text { Significance } F \\ \hline \text { Regression } & 2 & 9.421 & 4.710 & 1676.610 & 0.000 \\ \text { Residual } & 5 & 0.014 & 0.003 & & \\ \text { Total } & {7} & 9.435 & & & \\ \hline & \text { Coefficients } & \text { Standard Error } & t \text { Stat } & \text { P-value } & \\ \hline \text { Intercept } & 0.196 & 0.074 & 2.656 & 0.045 & \\ \text { x } & -0.100 & 0.038 & -2.652 & 0.045 & \\ \text { x-Sq } & 0.062 & 0.004 & 15.138 & 0.000 & \\ \hline \end{array} \end{aligned} $$ a. What percentage of the total variation is explained by the quadratic regression of \(y\) on \(x ?\) b. Is the regression on \(x\) and \(x^{2}\) significant at the \(\alpha=.05\) level of significance? c. Is the linear regression coefficient significant when \(x^{2}\) is in the model? d. Is the quadratic regression coefficient significant when \(x\) is in the model? e. The data were fitted to a linear model without the quadratic term with the results that follow. What can you say about the contribution of the quadratic term when it is included i \(\mathrm{n}\) the model? $$ \begin{array}{lc} \text { SUMMARY OUTPUT } & \\ \hline {\text { Regression Statistics }} \\ \hline \text { Multiple R } & 0.9645 \\ \text { R Square } & 0.9303 \\ \text { Adjusted R Square } & 0.9187 \\ \text { Standard Error } & 0.3311 \\ \text { Observations } & 8 \\ \hline \end{array} $$ $$ \begin{aligned} &\text { ANOVA }\\\ &\begin{array}{lccccc} \hline & d f & S S & M S & F & & \text { Significance } F \\ \hline \text { Regression } & 1 & 8.777 & 8.777 & 80.052 & 0.000 \\ \text { Residual } & 6 & 0.658 & 0.110 & & & \\ \text { Total } & 7 & 9.435 & & & & \\ \hline & & {\text { Standard }} \\ & \text { Coefficients } & \text { Error } & t \text { Stat } & P \text { -value } \\ \hline \text { Intercept } & -0.732 & 0.258 & -2.838 & 0.030 \\ x & 0.457 & 0.051 & 8.947 & 0.000 \\ \hline \end{array} \end{aligned} $$ f. The plot of the residuals from the linear regression model in part e shows a specific pattern. What is the term in the model that seems to be missing?

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In a study to examine the relationship between the time required to complete a construction project and several pertinent independent variables, an analyst compiled a list of four variables that might be useful in predicting the time to completion. These four variables were size of the contract, \(x_{1}\) (in $\$ 1000\( unit), number of workdays adversely affected by the weather \)x_{2},$ number of subcontractors involved in the project \(x_{4},\) and a variable \(x_{3}\) that measured the presence \(\left(x_{3}=1\right)\) or absence \(\left(x_{3}=0\right)\) of a workers' strike during the construction. Fifteen construction projects were randomly chosen, and each of the four variables as well as the time to completion were measured. $$ \begin{array}{rrrrrr} y & x_{1} & x_{2} & x_{3} & x_{4} \\ \hline 29 & 60 & 7 & 0 & 7 \\ 15 & 80 & 10 & 0 & 8 \\ 60 & 100 & 8 & 1 & 10 \\ 10 & 50 & 14 & 0 & 5 \\ 70 & 200 & 12 & 1 & 11 \\ 15 & 50 & 4 & 0 & 3 \\ 75 & 500 & 15 & 1 & 12 \\ 30 & 75 & 5 & 0 & 6 \\ 45 & 750 & 10 & 0 & 10 \\ 90 & 1200 & 20 & 1 & 12 \\ 7 & 70 & 5 & 0 & 3 \\ 21 & 80 & 3 & 0 & 6 \\ 28 & 300 & 8 & 0 & 8 \\ 50 & 2600 & 14 & 1 & 13 \\ 30 & 110 & 7 & 0 & 4 \end{array} $$ An analysis of these data using a first-order model in \(x_{1}\), $x_{2}, x_{3},\( and \)x_{4}$ produced the following printout. Give a complete analysis of the printout and interpret your results. What can you say about the apparent contribution of \(x_{1}\) and \(x_{2}\) in predicting \(y ?\) $$ \begin{array}{lr} \text { SUMMARY OUTPUT } & \\ \hline {\text { Regression Statistics }} \\ \hline \text { Multiple R } & 0.9204 \\ \text { R Square } & 0.8471 \\ \text { Adjusted R Square } & 0.7859 \\ \text { Standard Error } & 11.8450 \\ \text { Observations } & 15 \end{array} $$ $$ \begin{aligned} &\text { ANOVA }\\\ &\begin{array}{lccccc} \hline & d f & S S & M S & F & \text { Significance } F \\ \hline \text { Regression } & 4 & 7770.297 & 1942.574 & 13.846 & 0.000 \\ \text { Residual } & 10 & 1403.036 & 140.304 & & \\ \text { Total } & 14 & 9173.333 & & & \\ \hline \end{array} \end{aligned} $$ $$ \begin{array}{lcrll} & \text { Coefficients } & \text { Standard Error } & t \text { Stat } & P \text { -value } \\ \hline \text { Intercept } & -1.589 & 11.656 & -0.136 & 0.894 \\ \text { x1 } & -0.008 & 0.006 & -1.259 & 0.237 \\ \text { x2 } & 0.675 & 1.000 & 0.675 & 0.515 \\ \times 3 & 28.013 & 11.371 & 2.463 & 0.033 \\ \times 4 & 3.489 & 1.935 & 1.803 & 0.102 \\ \hline \end{array} $$

A multiple linear regression model involving one qualitative and one quantitative independent variable produced this prediction equation: $$ \hat{y}=12.6+.54 x_{1}-1.2 x_{1} x_{2}+3.9 x_{2}^{2} $$ a. Which of the two variables is the quantitative variable? Explain. b. If \(x_{1}\) can take only the values 0 or \(1,\) find the two possible prediction equations for this experiment. c. Graph the two equations found in part b. Compare the shapes of the two curves.

A quality control engineer is interested in predicting the strength of particle board \(y\) as a function of the size of the particles \(x_{1}\) and two types of bonding compounds. If the basic response is expected to be a quadratic function of particle size, write a linear model that incorporates the qualitative variable "bonding compound" into the predictor equation.
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