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Chapter 14: Problem 28

Why is it desirable to have the explanatory variables spread out to test a hypothesis regarding \(\beta_{1}\) or to construct confidence intervals about \(\beta_{1} ?\)

Short Answer

Expert verified
Spreading out explanatory variables improves the accuracy of hypothesis tests and confidence intervals for \(\beta_{1}\).

Step by step solution

01

Understand the role of explanatory variables

Explanatory variables provide the necessary data to explain variations in the dependent variable. When these variables are spread out, they cover a wide range of observations.
02

Impact on testing hypothesis regarding \(\beta_{1}\)

A broader range of explanatory variables increases the accuracy and reliability of hypothesis tests regarding \(\beta_{1}\). The spread-out data reduces multicollinearity and provides a clearer understanding of the relationship between variables.
03

Constructing confidence intervals about \(\beta_{1}\)

With a broad range of explanatory variables, confidence intervals about \(\beta_{1}\) become narrower and more precise. This improved precision enhances the ability to make reliable predictions and inferences.
04

Conclusion

In summary, spreading out the explanatory variables helps improve the quality of hypothesis testing and confidence intervals for \(\beta_{1}\), leading to more robust and accurate results.

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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 fundamental tool in statistics that allows us to make decisions or inferences about population parameters based on sample data. When testing a hypothesis regarding \(\beta_{1}\) in regression analysis, we are essentially checking whether there is a significant relationship between the explanatory variable and the dependent variable.
To effectively test such hypotheses, it is crucial to have explanatory variables that are well-spread out because:
  • A broad range of values allows for more variability, giving a clearer picture of how changes in the explanatory variable affect the dependent variable.
  • This spread improves the accuracy of test statistics, like the t-test for regression coefficients.
  • It helps in reducing errors, particularly type I errors (falsely rejecting a null hypothesis) and type II errors (falsely failing to reject a null hypothesis).
In summary, having explanatory variables spread out enhances our ability to test hypotheses accurately and draw meaningful conclusions.
confidence intervals
Confidence intervals provide a range of values within which we expect the true population parameter to lie, with a certain level of confidence (usually 95%). In regression analysis, constructing confidence intervals around the coefficient \(\beta_{1}\) is crucial for understanding the precision of our estimates.
When the explanatory variables are spread out:
  • The estimation of \(\beta_{1}\) becomes more accurate and reliable.
  • Wider spread leads to narrower confidence intervals, indicating higher precision in our estimates.
  • This precision aids in making better predictions and understanding the strength of the relationship between variables.
Therefore, a well-distributed set of explanatory variables contributes to more precise confidence intervals, which in turn allows for more trustworthy interpretations and inferences.
multicollinearity
Multicollinearity refers to a situation in regression analysis where two or more explanatory variables are highly linearly related. This can cause several problems:
  • It makes it difficult to determine the individual effect of each explanatory variable on the dependent variable.
  • Standard errors of the estimated coefficients increase, making hypothesis tests less reliable.
  • Confidence intervals become wider, indicating less precision in the coefficient estimates.
Knowing this, spreading out the explanatory variables helps reduce multicollinearity because:
  • Greater variability in the data lowers the correlations among explanatory variables.
  • Reduced multicollinearity means more stable and reliable estimates of regression coefficients.
In conclusion, managing multicollinearity by ensuring a wide range of explanatory variables is essential for accurate model estimation and reliable inference.

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Most popular questions from this chapter

More Age Estimation In the article "Bigger Teeth for Longer Life? Longevity and Molar Height in Two Roe Deer Populations" (Biology Letters [June, 2007\(]\) vol. 3 no. 3 \(268-270)\), researchers developed a model to predict the tooth height (in \(\mathrm{mm}\) ), \(y\), of roe deer based on their age, \(x_{1}\), gender, \(x_{2}(0=\) female \(, 1=\) male \(),\) and location, \(x_{3}\) (Trois Fontaines deer, which have a shorter life expectancy, and Chiz茅, which have a longer life expectancy, \(x_{3}=0\) for Trois Fontaines, \(x_{3}=1\) for Chiz茅). The model is $$ \hat{y}=7.790-0.382 x_{1}-0.587 x_{2}-0.925 x_{3}+0.091 x_{2} x_{3} $$ (a) What is the expected tooth length of a female roe deer who is 12 years old and lives in Trois Fontaines? (b) What is the expected tooth length of a male roe deer who is 8 years old and lives in Chiz茅? (c) What is the interaction term? What does the coefficient of the interaction term imply about tooth length?

For the data set below, use a partial \(F\) -test to determine whether the variables \(x_{1}\) and \(x_{2}\) do not significantly help to predict the response variable, \(y .\) Use the \(\alpha=0.05\) level of significance. $$ \begin{array}{ccccc|ccccc} x_{1} & x_{2} & x_{3} & x_{4} & y & x_{1} & x_{2} & x_{3} & x_{4} & y \\ \hline 24.9 & 66.3 & 13.5 & 3.7 & 59.8 & 41.1 & 83.5 & 9.7 & 21.8 & 84.6 \\ \hline 26.7 & 100.6 & 15.7 & 11.4 & 66.3 & 25.4 & 112.7 & 9.8 & 16.4 & 87.3 \\\ \hline 30.6 & 77.8 & 13.8 & 15.7 & 76.5 & 33.8 & 68.8 & 6.8 & 25.9 & 88.5 \\ \hline 39.6 & 83.4 & 8.8 & 8.8 & 77.1 & 23.5 & 69.5 & 7.5 & 15.5 & 90.7 \\ \hline 33.1 & 69.4 & 10.6 & 18.3 & 81.9 & 39.8 & 63.0 & 6.8 & 30.8 & 93.4 \\ \hline \end{array} $$

Life Cycle Hypothesis In the \(1950 \mathrm{~s}\), Franco Modigliani developed the Life Cycle Hypothesis. One tenet of this hypothesis is that income varies with age. The following data represent the annual income and age of a random sample of 15 adult Americans. $$ \begin{array}{cc|cc} \text { Age, } x & \text { Income, } y & \text { Age, } x & \text { Income, } y \\ \hline 25 & 25,490 & 47 & 41,398 \\ \hline 27 & 26,910 & 52 & 36,474 \\ \hline 32 & 32,141 & 54 & 38,934 \\ \hline 37 & 35,893 & 57 & 35,775 \\ \hline 42 & 36,451 & 62 & 30,629 \\ \hline 42 & 38,093 & 67 & 22,708 \\ \hline 47 & 36,266 & 72 & 20,506 \end{array} $$ (a) Draw a scatter diagram of the data. What type of relation appears to exist between \(x\) and \(y ?\) (b) Find the quadratic regression equation \(\hat{y}=b_{0}+b_{1} x+b_{2} x^{2}\). (c) Draw a residual plot against the fitted values, \(x,\) and \(x^{2}\). Also, draw a boxplot of the residuals. Are there any problems with the model? (d) Interpret the coefficient of determination. (e) Does the \(F\) -test indicate that we should reject \(H_{0}: \beta_{1}=\beta_{2}=0 ?\) Is either coefficient not significantly different from zero? (f) Construct and interpret \(95 \%\) confidence and prediction intervals incomes for an age of 45 years.

You obtain the multiple regression equation \(\hat{y}=5+3 x_{1}-4 x_{2}\) from a set of sample data. (a) Interpret the slope coefficients for \(x_{1}\) and \(x_{2}\). (b) Determine the regression equation with \(x_{1}=10\). Graph the regression equation with \(x_{1}=10\). (c) Determine the regression equation with \(x_{1}=15 .\) Graph the regression equation with \(x_{1}=15\) (d) Determine the regression equation with \(x_{1}=20 .\) Graph the regression equation with \(x_{1}=20 .\) (e) What is the effect of changing the value \(x_{1}\) on the graph of the regression equation?

(a) Draw a scatter diagram of the data. What type of relation appears to exist between \(x\) and \(y ?\) (b) Find the quadratic regression equation \(\hat{y}=b_{0}+b_{1} x+b_{2} x^{2}\) (c) Draw a residual plot against the fitted values, \(x,\) and \(x^{2}\). Also. draw a boxplot of the residuals. Are there any problems with the model? (d) Interpret the coefficient of determination. (e) Does the \(F\) -test indicate that we should reject \(H_{0}: \beta_{1}=\beta_{2}=0 ?\) Is either coefficient not significantly different from zero? (f) Construct and interpret \(95 \%\) confidence and prediction intervals for \(x=4\) $$ \begin{array}{cc} x & y \\ \hline 2.3 & 19.3 \\ \hline 2.7 & 14.8 \\ \hline 3.2 & 10.2 \\ \hline 4.1 & 4.8 \\ \hline 4.9 & 2.9 \\ \hline 5.6 & 3.9 \\ \hline 6.4 & 7.9 \\ \hline \end{array} $$
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