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Chapter 5: Problem 34

Explain why the slope \(b\) of the least-squares line always has the same sign (positive or negative) as does the sample correlation coefficient \(r\).

Short Answer

Expert verified
The slope of the least-squares line \(b\) always has the same sign as the sample correlation coefficient \(r\) because the slope \(b\) is calculated as \(r\) times the ratio of the standard deviations of \(y\) and \(x\). Since standard deviations are always non-negative, the sign of \(b\) will always match the sign of \(r\).

Step by step solution

01

Understand the correlation coefficient

The correlation coefficient \(r\) is calculated as \[r= \frac{{\sum{(x_i-\bar{x})(y_i-\bar{y})}}}{{\sqrt{\sum{(x_i-\bar{x})^2}\sum{(y_i-\bar{y})^2}}}}\] It measures the direction and strength of the linear relationship between two variables.
02

Understand the slope of the least squares line

The slope of the least squares line, \(b\), is calculated as \[b=r \cdot \frac{{s_y}}{{s_x}}\] where \(s_x\) and \(s_y\) represent the standard deviations of \(x\) and \(y\), respectively.
03

Relationship between the correlation coefficient and slope

From the above equations, it can be deduced that the sign of \(b\) is determined by the sign of \(r\). This is because \(s_x\) and \(s_y\) are standard deviations, which are always non-negative. Hence, the sign of \(b\) will match the sign of \(r\). If \(r\) is positive, \(b\) will be positive indicating a positive linear relationship. Conversely, if \(r\) is negative, \(b\) will also be negative indicating a negative linear relationship.

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Key Concepts

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

Least Squares Line
The least squares line is a concept frequently used in statistics to create the best possible straight line through a set of data points on a scatter plot. It's a critical concept because it helps in predicting values and understanding relationships between variables. The line minimizes the sum of the squares of the vertical distances (errors) between the actual data points and the points on the line itself.
This method ensures that the line is the best fit for the data in terms of overall deviation. Mathematically, it involves calculations of the slope \( b \) and the intercept. Here, the slope \( b \) is strongly connected to the correlation between the variables, which simplifies to \( b = r \cdot \frac{s_y}{s_x} \).
The least squares method relies heavily on solid mathematical grounding to efficiently draw insights from complex datasets and interpret relationships in a simplified linear manner.
Linear Relationship
A linear relationship is one where two variables move in a straight-line pattern. This implies a constant rate of increase or decrease in one variable when there is an increase or decrease in the other.
Such relationships can be perfectly positive, negative, or somewhere in between, and they are quantified using the correlation coefficient \( r \).
The correlation coefficient can take any value from \(-1\) to \(1\). A value of \(1\) indicates a perfect positive linear relationship, \(-1\) indicates a perfect negative relationship, and \(0\) suggests no linear relationship at all.
  • Positive \( r \): As one variable increases, the other tends to increase.
  • Negative \( r \): As one variable increases, the other tends to decrease.
The notion of linear relationships is fundamental in regression analysis, as it provides essential insights into how variables interact and pave the way for accurate predictions.
Standard Deviation
Standard deviation is a measure of the amount of variation or dispersion in a set of values. In statistics, understanding this concept is crucial because it gives insights into the spread and consistency of data points in a dataset.
A low standard deviation indicates that the data points tend to be close to the mean of the dataset, while a high standard deviation hints that the data points are spread out over a larger range of values.
Standard deviation is represented as \( s \), and when it's calculated for different datasets like \( x \) and \( y \), it forms the backbone of various statistical models including the least squares method. This mathematical tool helps adjust the scale in equations, ensuring that variables are comparably evaluated despite variations in their units or scales.
  • Useful for comparing varying datasets.
  • Helps in identifying the consistency of data points.
Understanding standard deviation is fundamental for anyone dealing with data, ensuring they can accurately interpret and analyze the spread and implications of statistical findings.

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Most popular questions from this chapter

A sample of 548 ethnically diverse students from Massachusetts were followed over a 19 -month period from 1995 and 1997 in a study of the relationship between TV viewing and eating habits (Pediatrics [2003]: \(1321-\) 1326). For each additional hour of television viewed per day, the number of fruit and vegetable servings per day was found to decrease on average by \(0.14\) serving. a. For this study, what is the dependent variable? What is the predictor variable? b. Would the least-squares line for predicting number of servings of fruits and vegetables using number of hours spent watching TV as a predictor have a positive or negative slope? Explain.

Cost-to-charge ratio (the percentage of the amount billed that represents the actual cost) for inpatient and outpatient services at 11 Oregon hospitals is shown in the following table (Oregon Department of Health Services, 2002). A scatterplot of the data is also shown. $$ \begin{array}{ccc} \hline \text { Hospital } & \begin{array}{l} \text { Outpatient } \\ \text { Care } \end{array} & \begin{array}{l} \text { Inpatient } \\ \text { Care } \end{array} \\ \hline 1 & 62 & 80 \\ 2 & 66 & 76 \\ 3 & 63 & 75 \\ 4 & 51 & 62 \\ 5 & 75 & 100 \\ 6 & 65 & 88 \\ 7 & 56 & 64 \\ 8 & 45 & 50 \\ 9 & 48 & 54 \\ 10 & 71 & 83 \\ 11 & 54 & 100 \\ \hline \end{array} $$ The least-squares regression line with \(y=\) inpatient costto-charge ratio and \(x=\) outpatient cost-to-charge ratio is \(\hat{y}=-1.1+1.29 x\). a. Is the observation for Hospital 11 an influential observation? Justify your answer. b. Is the observation for Hospital 11 an outlier? Explain. c. Is the observation for Hospital 5 an influential observation? Justify your answer. d. Is the observation for Hospital 5 an outlier? Explain.

The accompanying data on \(x=\) head circumference \(z\) score (a comparison score with peers of the same age - a positive score suggests a larger size than for peers) at age 6 to 14 months and \(y=\) volume of cerebral grey matter (in ml) at age 2 to 5 years were read from a graph in the article described in the chapter introduction (Journal of the American Medical Association [2003]). $$ \begin{array}{cc} & \text { Head Circumfer- } \\ \text { Cerebral Grey } & \text { ence } z \text { Scores at } \\ \text { Matter (ml) 2-5 yr } & \text { 6-14 Months } \\ \hline 680 & -.75 \\ 690 & 1.2 \\ 700 & -.3 \\ 720 & .25 \\ 740 & .3 \\ 740 & 1.5 \\ 750 & 1.1 \\ 750 & 2.0 \\ 760 & 1.1 \\ 780 & 1.1 \\ 790 & 2.0 \\ 810 & 2.1 \\ 815 & 2.8 \\ 820 & 2.2 \\ 825 & .9 \\ 835 & 2.35 \\ 840 & 2.3 \\ 845 & 2.2 \\ \hline \end{array} $$ a. Construct a scatterplot for these data. b. What is the value of the correlation coefficient? c. Find the equation of the least-squares line. d. Predict the volume of cerebral grey matter for a child whose head circumference \(z\) score at age 12 months was \(1.8\). e. Explain why it would not be a good idea to use the least-squares line to predict the volume of grey matter for a child whose head circumference \(z\) score was \(3.0\).

In the study of textiles and fabrics, the strength of a fabric is a very important consideration. Suppose that a significant number of swatches of a certain fabric are subjected to different "loads" or forces applied to the fabric. The data from such an experiment might look as follows: $$ \begin{aligned} &\text { Hypothetical Data on Fabric Strength }\\\ &\begin{array}{lccccccc} \hline \begin{array}{l} \text { Load } \\ \text { (lb/sq in.) } \end{array} & \mathbf{5} & \mathbf{1 5} & \mathbf{3 5} & \mathbf{5 0} & \mathbf{7 0} & \mathbf{8 0} & \mathbf{9 0} \\ \hline \begin{array}{l} \text { Proportion } \\ \text { failing } \end{array} & 0.02 & 0.04 & 0.20 & 0.23 & 0.32 & 0.34 & 0.43 \\ \hline \end{array} \end{aligned} $$ a. Make a scatterplot of the proportion failing versus the load on the fabric. b. Using the techniques introduced in this section, calculate \(y^{\prime}=\ln \left(\frac{p}{1-p}\right)\) for each of the loads and fit the line \(y^{\prime}=a+b\) (Load). What is the significance of a positive slope for this line? c. What proportion of the time would you estimate this fabric would fail if a load of \(60 \mathrm{lb} / \mathrm{sq}\) in. were applied? d. In order to avoid a "wardrobe malfunction," one would like to use fabric that has less than a \(5 \%\) chance of failing. Suppose that this fabric is our choice for a new shirt. To have less than a \(5 \%\) chance of failing, what would you estimate to be the maximum "safe" load in \(\mathrm{lb} / \mathrm{sq}\) in.?

The accompanying data represent \(x=\) the amount of catalyst added to accelerate a chemical reaction and \(y=\) the resulting reaction time: $$ \begin{array}{rrrrrr} x & 1 & 2 & 3 & 4 & 5 \\ y & 49 & 46 & 41 & 34 & 25 \end{array} $$ a. Calculate \(r\). Does the value of \(r\) suggest a strong linear relationship? b. Construct a scatterplot. From the plot, does the word linear really provide the most effective description of the relationship between \(x\) and \(y\) ? Explain.
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