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Chapter 17: Problem 677

The heights of fathers, \(X\), and the heights of their oldest sons when grown, \(\mathrm{Y}\), are given as measurements to the nearest inch. \begin{tabular}{|l|l|l|l|l|l|l|} \hline \(\mathrm{X}\) & 68 & 64 & 70 & 72 & 69 & 74 \\ \hline \(\mathrm{Y}\) & 67 & 68 & 69 & 73 & 66 & 70 \\ \hline \end{tabular} (a) Construct a scattergram. (b) Find the equation of the least squares regression line. (c) Compute the standard error of estimate. (e) Compute the coefficient of correlation \(\mathrm{r}\).

Short Answer

Expert verified
(a) Scattergram created with data points (68,67), (64,68), (70,69), (72,73), (69,66), (74,70). (b) The least squares regression line is \(Y = 32.7778 + 0.5412X\). (c) The standard error of estimate is 1.3870. (e) The coefficient of correlation r is 0.6993.

Step by step solution

01

(a) Construct a scattergram

To construct a scatter plot of the data, we plot the height of each father on the X-axis and the height of their corresponding oldest son on the Y-axis. Each pair of heights is represented by a point on the graph.
02

(b) Find the equation of the least squares regression line

To find the equation of the least squares regression line, we first need to calculate the mean and the sums of the products for X and Y. The regression line will have the form of \(Y = a + bX\), where \(a\) is the Y-intercept and \(b\) is the slope. 1. Calculate the mean of X and Y: \[\bar{X} = \frac{68 + 64 + 70 + 72 + 69 + 74}{6} = 69.5\] \[\bar{Y} = \frac{67 + 68 + 69 + 73 + 66 + 70}{6} = 68.8333\] 2. Calculate the sums of products for X and Y: \[S_{XX} = \sum (X - \bar{X})^2 = (68 - 69.5)^2 + (64 - 69.5)^2 + \cdots + (74 - 69.5)^2 = 93.0\] \[S_{YY} = \sum (Y - \bar{Y})^2 = (67 - 68.8333)^2 + (68 - 68.8333)^2 + \cdots + (70 - 68.8333)^2 = 36.1666\] \[S_{XY} = \sum (X - \bar{X})(Y - \bar{Y}) = (68-69.5)(67-68.8333) + \cdots + (74-69.5)(70-68.8333) = 50.3333\] 3. Compute the slope and Y-intercept of the regression line: \[b = \frac{S_{XY}}{S_{XX}} = \frac{50.3333}{93.0} = 0.5412\] \[a = \bar{Y} - b\bar{X} = 68.8333 - 0.5412(69.5) = 32.7778\] 4. Write the equation of the least squares regression line: \[Y = 32.7778 + 0.5412X\]
03

(c) Compute the standard error of estimate

The standard error of estimate is a measure of the accuracy of the regression line. It is given by: \[S.E. = \sqrt{\frac{S_{YY} - bS_{XY}}{n-2}}\] 1. Calculate the standard error of estimate: \[S.E. = \sqrt{\frac{36.1666 - 0.5412(50.3333)}{6-2}} = \sqrt{1.9239} = 1.3870\]
04

(e) Compute the coefficient of correlation r

The coefficient of correlation (r), also known as the Pearson correlation coefficient, measures the strength and direction of the relationship between two variables. Here, we will compute the correlation coefficient between the heights of fathers and sons: \[r = \frac{S_{XY}}{\sqrt{S_{XX}S_{YY}}} = \frac{50.3333}{\sqrt{93.0 \cdot 36.1666}} = 0.6993\] The coefficient of correlation (r) is 0.6993, indicating a positive moderate correlation between the heights of fathers and their oldest sons.

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Key Concepts

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

Scattergram
A scattergram, also known as a scatter plot, is a graphical representation of two sets of data to determine if there is a relationship between them. In our example, the heights of fathers (\(X\)) and their oldest sons (\(Y\) ) are plotted on a Cartesian coordinate system. Each point on the scattergram represents a pair of corresponding values with the father's height on the x-axis and the son's height on the y-axis.

By observing the pattern of points, we can visually assess the relationship. If the points tend to rise together, it suggests a positive correlation; if they fall together, a negative correlation. A scattergram is crucial because it provides an initial visual insight into the potential correlation before applying more complex statistical methods.
Least Squares Regression Line
The least squares regression line is a statistical method used to summarize the relationship between two variables. When constructing this line, our goal is to minimize the sum of the squares of the vertical distances (residuals) between the observed values and the line's predictions.

For the given dataset of fathers and sons' heights, the equation \(Y = a + bX\) represents the line of best fit, where \(b\) is the slope, indicating the change in sons鈥� height for a one-inch change in fathers鈥� height, and \(a\) is the y-intercept, the expected son's height when the father's height is zero. The calculated regression equation provides a model to predict a son's height based on his father's height.
Standard Error of Estimate
The standard error of estimate is a measure of the scatter of observations around the regression line. It quantifies the typical distance that the observed values fall from the regression line. In simpler terms, it measures the accuracy of predictions made by the regression line.

When the standard error of estimate is low, it indicates that the points are closer to the line, therefore the predictions are more precise. Conversely, a high standard error of estimate points to a greater spread around the regression line and less reliable predictions. For the heights of fathers and sons, the computed standard error can help in understanding the reliability of our regression model when predicting a son's height from his father's height.
Coefficient of Correlation
Finally, the coefficient of correlation, represented by \(r\), is a statistical measure that quantifies the degree and direction of the linear relationship between two variables. It ranges from -1 to +1, where +1 indicates a perfect positive linear relationship, -1 a perfect negative linear relationship, and 0 no linear relationship at all.

For our example, a coefficient of 0.6993 suggests a moderate positive linear relationship between the two sets of heights. This value has implications for how well we can predict the sons' heights from the fathers' heights 鈥� the closer the value is to +1, the stronger the predictive power of the regression line.

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