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Chapter 13: Problem 86

Briefly explain the difference between estimating the mean value of \(y\) and predicting a particular value of \(y\) using a regression model.

Short Answer

Expert verified
Estimating the mean value of \(y\) in a regression model refers to calculating the average response or the typical value of the dependent variable, usually done by taking the average value of \(x\). But predicting a specific value of \(y\) means finding the exact \(y\) outcome for a certain \(x\) value, considering both the average response and individual variability.

Step by step solution

01

Understand the Concept of Estimating Mean in Regression

Estimating the mean of \(y\) refers to calculating the average value of the dependent variable \(y\). This is typically done by using the average value of the independent variable \(x\), referred to as \(E(y|x)\) in a regression model. The mean is a central measure that shows the point where data is centered or clustered around in a data set.
02

Understand the Concept of Predicting a Value in Regression

Predicting a specific value of \(y\) denotes trying to find the exact \(y\) value for a particular \(x\) value using the regression equation built from the data. The prediction is a specific point estimate that shows the expected value of \(y\) for a given value of \(x\).
03

Explanation of the Difference between Both Concepts

When you're estimating the mean value of \(y\), you're essentially looking for an overall, average \(y\) response to a change in \(x\), while ignoring random variability. However, when predicting a specific \(y\) value, individual variability or noise become significant factors because the intention here is to predict a specific result given the specific conditions.

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Most popular questions from this chapter

Briefly explain the difference between a deterministic and a probabilistic regression model.

The following table lists the midterm and final exam scores for seven students in a statistics class. $$ \begin{array}{l|lllllll} \hline \text { Midterm score } & 79 & 95 & 81 & 66 & 87 & 94 & 59 \\ \hline \text { Final exam score } & 85 & 97 & 78 & 76 & 94 & 84 & 67 \\ \hline \end{array} $$ a. Do you expect the midterm and final exam scores to be positively or negatively related? b. Plot a scatter diagram. By looking at the scatter diagram, do you expect the correlation coefficient between these two variables to be close to zero, 1 , or \(-1\) ? c. Find the correlation coefficient. Is the value of \(r\) consistent with what you expected in parts a and \(\mathrm{b}\) ? d. Using the \(1 \%\) significance level, test whether the linear correlation coefficient is positive.

Explain the meaning of independent and dependent variables for a regression model.

Two variables \(x\) and \(y\) have a negative linear relationship. Explain what happens to the value of \(y\) when \(x\) increases.

The health department of a large city has developed an air pollution index that measures the level of several air pollutants that cause respiratory distress in humans. The accompanying table gives the pollution index (on a scale of 1 to 10 , with 10 being the worst) for 7 randomly selected summer days and the number of patients with acute respiratory problems admitted to the emergency rooms of the city's hospitals. $$ \begin{array}{l|ccccccc} \hline \text { Air pollution index } & 4.5 & 6.7 & 8.2 & 5.0 & 4.6 & 6.1 & 3.0 \\\ \hline \text { Emergency admissions } & 53 & 82 & 102 & 60 & 39 & 42 & 27 \\ \hline \end{array} $$ a. Taking the air pollution index as an independent variable and the number of emergency admissions as a dependent variable, do you expect \(B\) to be positive or negative in the regression model \(y=A+B x+\epsilon ?\) b. Find the least squares regression line. Is the sign of \(b\) the same as you hypothesized for \(B\) in part a? c. Compute \(r\) and \(r^{2}\), and explain what they mean. d. Compute the standard deviation of errors. e. Construct a \(90 \%\) confidence interval for \(B\). f. Test at the \(5 \%\) significance level whether \(B\) is positive. g. Test at the \(5 \%\) significance level whether \(\rho\) is positive. Is your conclusion the same as in part \(\mathrm{f}\) ?
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