91Ó°ÊÓ

With a bit of algebra, we can show that $$ \text { SSResid }=\left(1-r^{2}\right) \sum(y-\bar{y})^{2} $$ from which it follows that $$ s_{e}=\sqrt{\frac{n-1}{n-2}} \sqrt{1-r^{2}} s_{y} $$ Unless \(n\) is quite small, \((n-1) /(n-2) \approx 1\), so $$ s_{e} \approx \sqrt{1-r^{2}} s_{y} $$ a. For what value of \(r\) is \(s_{e}\) as large as \(s_{y}\) ? What is the least- squares line in this case? b. For what values of \(r\) will \(s_{e}\) be much smaller than \(s_{s} ?\) c. A study by the Berkeley Institute of Human Development (see the book Statistics by Freedman et al. listed in the back of the book) reported the following summary data for a sample of \(n=66\) California boys: \(r \approx .80\) At age 6 , average height \(\approx 46\) inches, standard deviation \(\approx 1.7\) inches. At age 18 , average height \(\approx 70\) inches, standard deviation \(\approx 2.5\) inches. What would \(s_{e}\) be for the least-squares line used to predict 18 -year-old height from 6 -year-old height? d. Referring to Part (c), suppose that you wanted to predict the past value of 6 -year-old height from knowledge of 18 -year-old height. Find the equation for the appropriate least-squares line. What is the corresponding value of \(\left.s_{e}\right\\}\)

Step by step solution

01

(a) Value for which \(s_{e} = s_{y}\)

The first step is intended to find the correlation coefficient (\(r\)) for which the Residual Standard Deviation (\(s_{e}\)) equals the standard deviation of \(y\) (\(s_{y}\)). According to the formula \(s_{e} \approx \sqrt{1-r^{2}} s_{y}\), equate \(s_{e}\) and \(s_{y}\) and solve for \(r\). The solution is \(r = 0\)

02

(a) Least squares line

The least-squares line for a correlation coefficient of 0 indicates no linear relationship. In this case, the best prediction for all values of \(y\) is their mean value. Therefore, the equation is \(y = \overline{y}\)

03

(b) Value for which \(s_{e}\) is much smaller than \(s_{y}\)

In this step, identify values of the correlation coefficient (\(r\)) for which the Residual Standard Deviation (\(s_{e}\)) is much smaller than the standard deviation of \(y\) (\(s_{y}\)). Considering the formula \(s_{e} \approx \sqrt{1-r^{2}} s_{y}\), it's clear that the deviation \(s_{e}\) will be much smaller than \(s_{y}\) for values of \(r\) close to \(\pm 1\). This is because for \(|r| \approx 1\), \(\sqrt{1-r^2} \approx 0\), meaning \(s_{e}\) will be substantially less than \(s_{y}\).

04

(c) Calculate \(s_{e}\) for 18-year-old height predictions

Using the provided data, \(r = 0.8\) and \(s_{y} = 2.5\) inches, calculate \(s_{e}\) using the formula \(s_{e} \approx \sqrt{1-r^{2}} s_{y}\). This equals approximately 1.5 inches.

05

(d) Determine least-squares line for predicting past value

The least-squares line for predicting the 6-year-old height from the 18-year-old height has the same structure as the standard form \(y = mx + c\), where the slope \(m\) is the change in 6-year-old height per unit change in 18-year-old height, and \(c\) is the intercept. Solve the mentioned equation using the given values of averages and standard deviations.

06

(d) Calculate corresponding \(s_{e}\) value

Since the correlation coefficient \(r\) is invariant to whether \(y\) are the 6-year-old or 18-year-old heights, you can use the previously computed value of \(r\), but with the standard deviation of 6-year-old heights (\(s_{y} = 1.7\)). Substituting the above values in the formula used above, compute the new \(s_{e}\) value for predicting a 6-year-old's height from an 18-year-old's height

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Key Concepts

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

Correlation Coefficient

Understanding the correlation coefficient is crucial when analyzing relationships between variables in statistics. It is denoted as \( r \) and measures the strength and direction of a linear relationship between two variables on a scatterplot. The value of \( r \) ranges from -1 to 1.

A correlation coefficient close to 1 implies a strong positive linear relationship, indicating that as one variable increases, so does the other. Conversely, a coefficient close to -1 indicates a strong negative linear relationship, showing an inverse association between the variables. A correlation coefficient around 0 implies little to no linear relationship between the variables.

In the context of residual standard deviation, the correlation coefficient influences \( s_{e} \), the estimate of the standard deviation of the residuals from the least-squares line. As the coefficient moves further from 0 towards \( \pm 1 \), the smaller \( s_{e} \) becomes in comparison to the standard deviation of the original data points, \( s_{y} \). This suggests that the predictions from the least-squares line are more accurate as the correlation increases in magnitude.

Least-Squares Line

The least-squares line, also known as the line of best fit, is a fundamental concept in regression analysis used to model the relationship between two variables. The goal of the least-squares line is to minimize the sum of the squared differences (residuals) between the observed values and the values predicted by the line.

To construct the least-squares line, we apply the least-squares method, which involves the use of calculus to find the line that minimizes the sum of the squares of the vertical distances of the points from the line. The line is mathematically expressed as \( y = mx + b \), where \( m \) is the slope and \( b \) the y-intercept. The slope is calculated based on the correlation between the variables and their standard deviations, while the intercept takes into account their means.

When the correlation coefficient is zero, the least-squares line is simply a horizontal line at the mean value of the dependent variable, indicating no predictive relationship. As the absolute value of the correlation coefficient increases, the line more accurately represents the data, resulting in a lower residual standard deviation.

Predictive Analytics in Statistics

Predictive analytics in statistics involves using historical data, statistical algorithms, and machine learning techniques to make predictions about future or otherwise unknown events. The heart of predictive analytics lies in the models built to forecast outcomes, and the least-squares regression line is one of these models.

In predictive analytics, the correlation coefficient and the least-squares line are crucial for building reliable predictive models. A higher correlation coefficient indicates a stronger relationship between variables, leading to a more significant predictive power. The least-squares line becomes a predictive model that can infer future trends or estimate unknown values.

By analyzing the residuals, which are the differences between observed and predicted values, statisticians refine their models for better accuracy. The residual standard deviation helps assess the variability in these predictions, and by minimizing \( s_{e} \), the predictions become more reliable. The predictive analytics process is highly iterative, involving model building, testing, validation, and refinement to improve forecasts.