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Chapter 14: Problem 1

Suppose a least-squares regression line is given by \(\hat{y}=4.302 x-3.293 .\) What is the mean value of the response variable if \(x=20 ?\)

Short Answer

Expert verified
The mean value of the response variable is 82.747 when \( x = 20 \).

Step by step solution

01

- Identify the given regression equation

The given least-squares regression line is \[\hat{y} = 4.302x - 3.293\].
02

- Substitute the given value of x into the equation

We are given that \( x = 20 \). Substitute \( x = 20 \) into the equation: \[\hat{y} = 4.302(20) - 3.293\].
03

- Compute the product and subtraction

First, compute the product: \(4.302 \times 20 = 86.04\).Then, subtract 3.293 from 86.04: \(86.04 - 3.293 = 82.747\).

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Key Concepts

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

regression analysis
Regression analysis is a crucial statistical method used to understand the relationship between two or more variables. A common form is the least-squares regression line, which helps us make predictions based on that relationship.
In our example, the regression equation \(\text{\hat{y}} = 4.302x - 3.293\) relates variable \(x\) to the response variable \(\hat{y}\). We use it to predict \(\hat{y}\) for any given value of \(x\).
Such an equation is derived by minimizing the sum of the squares of the differences between the observed values and the values predicted by the linear model.
This technique ensures that our predictions are as accurate as possible, given the data.
response variable
In regression analysis, the response variable, often denoted as \(\hat{y}\), is what we aim to predict or explain. It's also called the dependent variable because it depends on the input variable \(x\) (the independent variable).
For example, in our exercise, the response variable \(\hat{y}\) changes based on the value of \(x\). We used \(x = 20\) to find the corresponding response variable:
  • First we identified the regression equation: \(\text{\hat{y}} = 4.302x - 3.293\).
  • Substituting \(x = 20\), we calculated \(\hat{y}\) as 82.747 after evaluating the equation: \(4.302(20) - 3.293\).

Understanding the response variable is essential for analyzing how different inputs affect our outcomes.
mean value
The mean value refers to the average value of a set of numbers. In regression analysis, it's often used to interpret the central tendency of the response variable.
When we estimate the response variable using the regression equation, we're essentially finding its expected mean value for a specific \(x\).
For instance, with \(x = 20\), we calculated the mean value of \(\hat{y}\) by substituting into our regression formula: \(\text{\hat{y}} = 4.302(20) - 3.293\).
This gives us \(\hat{y} = 82.747\), which represents the mean predicted value of the response variable when \(x\) is 20.
Understanding the mean value helps in summarizing the data and recognizing trends within the data set.

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Most popular questions from this chapter

For the data set $$ \begin{array}{ccccc} \boldsymbol{x}_{1} & \boldsymbol{x}_{2} & \boldsymbol{x}_{3} & \boldsymbol{x}_{4} & \boldsymbol{y} \\ \hline 47.3 & 0.9 & 4 & 76 & 105.5 \\ \hline 53.1 & 0.8 & 6 & 55 & 113.8 \\ \hline 56.7 & 0.8 & 4 & 65 & 115.2 \\ \hline 48.8 & 0.5 & 7 & 67 & 118.9 \\ \hline 42.7 & 1.1 & 7 & 74 & 148.9 \\ \hline 44.3 & 1.1 & 6 & 76 & 120.2 \\ \hline 44.5 & 0.7 & 8 & 68 & 121.6 \\ \hline 37.7 & 0.7 & 7 & 79 & 140.0 \\ \hline 36.9 & 1.0 & 5 & 73 & 141.5 \\ \hline 28.1 & 1.8 & 6 & 68 & 141.9 \\ \hline 32.0 & 0.8 & 8 & 81 & 152.8 \\ \hline 34.7 & 0.8 & 10 & 68 & 156.5 \\\\\hline \end{array} $$ (a) Construct a correlation matrix between \(x_{1}, x_{2}, x_{3}, x_{4},\) and \(y\). Is there any evidence that multicollinearity may be a problem? (b) Determine the multiple regression line using all the explanatory variables listed. Does the \(F\) -test indicate that we should reject \(H_{0}: \beta_{1}=\beta_{2}=\beta_{3}=\beta_{4}=0 ?\) Which explanatory variables have slope coefficients that are not significantly different from zero? (c) Remove the explanatory variable with the highest \(P\) -value from the model and recompute the regression model. Does the \(F\) -test still indicate that the model is significant? Remove any additional explanatory variables on the basis of the \(P\) -value of the slope coefficient. Then compute the model with the variable removed. (d) Draw residual plots and a box plot of the residuals to assess the adequacy of the model. (e) Use the final model constructed in part (c) to predict the value of \(y\) if \(x_{1}=44.3, x_{2}=1.1, x_{3}=7,\) and \(x_{4}=69 .\) (f) Draw a normal probability plot of the residuals. Is it reasonable to construct confidence and prediction intervals? (g) Construct \(95 \%\) confidence and prediction intervals if \(x_{1}=44.3, x_{2}=1.1, x_{3}=7,\) and \(x_{4}=69 .\)

Divorce Rates The given data represent the percentage, \(y,\) of the population that is divorced for various ages, \(x\), in the United States in 2010 based on sample data obtained from the United States Statistical Abstract in \(2012 .\) $$ \begin{array}{cc} \text { Age, } x & \text { Percentage Divorced, } y \\ \hline 22 & 0.9 \\ \hline 27 & 3.6 \\ \hline 32 & 7.4 \\ \hline 37 & 10.4 \\ \hline 42 & 12.7 \\ \hline 50 & 15.7 \\ \hline 60 & 16.2 \\ \hline 70 & 13.1 \\ \hline 80 & 6.5 \end{array} $$ (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 a \(95 \%\) confidence interval for percent divorced among all 30 years olds.

(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 \(x=4\) $$ \begin{array}{lc} \boldsymbol{x} & \boldsymbol{y} \\ \hline 1 & 25.1 \\ \hline 1.5 & 19.3 \\ \hline 2.1 & 13.9 \\ \hline 3.6 & 8.5 \\ \hline 4.3 & 7.7 \\ \hline 5.9 & 12.9 \\ \hline 6.2 & 14.3 \\ \hline \end{array} $$

A researcher wants to determine a model that can be used to predict the 28 -day strength of a concrete mixture. The following data represent the 28 -day and 7 -day strength (in pounds per square inch) of a certain type of concrete along with the concrete's slump. Slump is a measure of the uniformity of the concrete, with a higher slump indicating a less uniform mixture. $$ \begin{array}{ccc} \text { Slump (inches) } & \text { 7-Day psi } & \text { 28-Day psi } \\ \hline 4.5 & 2330 & 4025 \\ \hline 4.25 & 2640 & 4535 \\\ \hline 3 & 3360 & 4985 \\ \hline 4 & 1770 & 3890 \\ \hline 3.75 & 2590 & 3810 \\ \hline 2.5 & 3080 & 4685 \\ \hline 4 & 2050 & 3765 \\ \hline 5 & 2220 & 3350 \\ \hline 4.5 & 2240 & 3610 \\ \hline 5 & 2510 & 3875 \\ \hline 2.5 & 2250 & 4475 \end{array} $$ (a) Construct a correlation matrix between slump, 7 -day psi, and 28 -day psi. Is there any reason to be concerned with multicollinearity based on the correlation matrix? (b) Find the least-squares regression equation \(\hat{y}=b_{0}+b_{1} x_{1}+b_{2} x_{2},\) where \(x_{1}\) is slump, \(x_{2}\) is 7 -day strength, and \(y\) is the response variable, 28 -day strength. (c) Draw residual plots and a boxplot of the residuals to assess the adequacy of the model. (d) Interpret the regression coefficients for the least-squares regression equation. (e) Determine and interpret \(R^{2}\) and the adjusted \(R^{2}\). (f) Test \(H_{0}: \beta_{1}=\beta_{2}=0\) versus \(H_{1}:\) at least one of the \(\beta_{1} \neq 0\) at the \(\alpha=0.05\) level of significance. (g) Test the hypotheses \(H_{0}: \beta_{1}=0\) versus \(H_{1}: \beta_{1} \neq 0\) and \(H_{0}: \beta_{2}=0\) versus \(H_{1}: \beta_{2} \neq 0\) at the \(\alpha=0.05\) level of significance. (h) Predict the mean 28 -day strength of all concrete for which slump is 3.5 inches and 7 -day strength is 2450 psi. (i) Predict the 28 -day strength of a specific sample of concrete for which slump is 3.5 inches and 7 -day strength is 2450 psi. (j) Construct \(95 \%\) confidence and prediction intervals for concrete for which slump is 3.5 inches and 7 -day strength is 2450 psi. Interpret the results.

Researchers developed a model to explain the age gap between husbands and wives at first marriage. The model is below: $$ \hat{y}=0.0321 x_{1}+0.9848 x_{2}+0.5391 x_{3}-0.000145 x_{4}^{2}+3.8483 $$ where y: Age gap at first marriage (male - female) \(x_{1}:\) Percent of children aged 10 to 14 involved in child labor \(x_{2}:\) Indicator variable where 1 is an African country, 0 otherwise \(x_{3}:\) Percent of the population that is Muslim \(x_{4}:\) Percent of the population that is literate Source: Xu Zhang and Solomon W. Polachek, State University of New York at Binghamton "The Husband- Wife Age Gap at First Marriage: A Cross-Country Analysis" (a) Use the model to predict the age gap at first marriage for an African country where the percent of children aged 10 to 14 who are involved in child labor is \(12,\) the percent of the population that is Muslim is \(30,\) and the percent of the population that is literate is \(75 .\) (b) What would be the mean difference in age gap between an African country and a non-African country? (c) Interpret the coefficient of "percent of children aged 10 to 14 involved in child labor." (d) The coefficient of determination for this model is 0.593 . Interpret this result. (e) The \(P\) -value for the test \(H_{1}: \beta_{1} \neq 0\) versus \(H_{1}: \beta_{1} \neq 0\) is \(0.008 .\) What would you conclude about this test?
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