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Chapter 5: Problem 33

The sales manager of a large company selected a random sample of \(n=10\) salespeople and determined for each one the values of \(x=\) years of sales experience and \(y=\) annual sales (in thousands of dollars). A scatterplot of the resulting \((x, y)\) pairs showed a marked linear pattern. a. Suppose that the sample correlation coefficient is \(r=\) \(.75\) and that the average annual sales is \(\bar{y}=100\). If a particular salesperson is 2 standard deviations above the mean in terms of experience, what would you predict for that person's annual sales? b. If a particular person whose sales experience is \(1.5\) standard deviations below the average experience is predicted to have an annual sales value that is 1 standard deviation below the average annual sales, what is the value of \(r ?\)

Short Answer

Expert verified
A salesperson with 2 standard deviations above the mean experience is predicted to have annual sales of $101.5$ thousand dollars. If a person whose sales experience is 1.5 standard deviations below the mean is predicted to have a sales value that is 1 standard deviation below the mean, the correlation value or \(r\) would be 1.5.

Step by step solution

01

Understand Prediction formula

For a linear regression model, the formula to predict \(y\) based on \(x\) is \(\hat{y} = \bar{y} + r \cdot (s_x / s_y) \cdot (x - \bar{x})\). Here, \(\hat{y}\) is the predicted value, \(\bar{y}\) is the average of \(y\), \(r\) is the correlation, \(s_x\) and \(s_y\) are standard deviations of \(x\) and \(y\), and \(x\) is the given data point.
02

Substitute Known Values to Predict Sales

In part a, it is said that a salesperson is 2 standard deviations above the mean in terms of experience which means \(x - \bar{x} = 2\), \( \bar{y}=100\), and \(r=0.75\). No concrete values were given for the standard deviation of \(x\) and \(y\), but the standard deviations will cancel out in the prediction formula because the difference of \(x\) from its mean is given in terms of standard deviations. Substituting these values in, the predicted annual sales become: \(\hat{y} = 100 + 0.75 \cdot 2 = 100 + 1.5 = 101.5\) thousand dollars.
03

Understand Regression Coefficient formula

The Regression Coefficient, \(r\), can be calculated using the formula \(r=\frac{s_{xy}}{s_{x}s_{y}}\), where \(s_{xy}\) is the covariance of \(x\) and \(y\), and \(s_{x}\) and \(s_{y}\) are the standard deviations of \(x\) and \(y\). In part b, the covariance of \(x\) and \(y\) is equal to their standard deviations, since the person's experience and sales both vary from the average in terms of standard deviations.
04

Substitute Known Values to Calculate Correlation

A salesperson is 1.5 standard deviations below the average in experience (\(-1.5 = x - \bar{x}\)), and the predicted annual sales value is 1 standard deviation below the average (\(-1 = \hat{y} - \bar{y}\)), thus, substituting these changes into the Regression Coefficient formula will give us the correlation \(r = -1.5/-1 = 1.5\).
05

Final Interpretation

A salesperson with 2 standard deviations above the mean experience is predicted to have an annual sales of 101.5 thousand dollars, given that \(r=0.75\). On the other hand, if a person with 1.5 standard deviations below the mean experience is predicted with 1 standard deviation below the average annual sales, the correlation would be 1.5.

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Key Concepts

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

Linear Regression
Linear regression is a powerful statistical method used to examine the relationship between two quantitative variables. It helps us predict the value of one variable based on the value of another. This is especially useful when the data points show a linear pattern, which means they follow a straight-line trend.

In a linear regression model, the formula to predict the value of the dependent variable (\( y \)) based on the independent variable (\( x \)) is given by:\[\hat{y} = \bar{y} + r \cdot \left(\frac{s_x}{s_y}\right) \cdot (x - \bar{x})\]where:
  • \( \hat{y} \): predicted value of \( y \)
  • \( \bar{y} \): average value of \( y \)
  • \( r \): correlation coefficient
  • \( s_x \) and \( s_y \): standard deviations of \( x \) and \( y \)
  • \( x - \bar{x} \): how far \( x \) is from its mean \( \bar{x} \)
Linear regression helps to make predictions and understand the strength and direction of the linear relationship between two variables.
Sample Data Analysis
Sample data analysis involves examining a subset of data collected from a larger population. This technique is commonly used because analyzing the entire population is often impractical or impossible. The goal is to draw conclusions and make inferences about the population based on this sample.

In the context of our exercise, a sales manager selected a random sample of 10 salespeople to examine the relationship between their years of experience and their annual sales. By analyzing this sample, we can estimate trends and make predictions for the entire group, such as predicting annual sales based on years of experience.

Using sample data helps in discovering patterns and correlations. It allows us to use techniques like linear regression to provide insights into potential future outcomes, even when we're working with a small portion of the data.
Scatterplots
Scatterplots are valuable visual tools that help us explore the relationship between two quantitative variables. By plotting each data point on a grid, where one variable is represented on the x-axis and the other on the y-axis, we can easily see patterns and trends.

In the context of our problem, a scatterplot was used to show the relationship between the years of experience of salespeople and their annual sales. If the points on the scatterplot tend to align along a path or show a general direction (upward or downward), this suggests a linear relationship.

Key insights from scatterplots:
  • Pattern recognition: Visualize whether there's a linear or non-linear pattern.
  • Direction: Identify if the relationship is positive or negative.
  • Strength: Assess how closely the points follow a straight line.
Scatterplots are essential for performing linear regression as they visually confirm the linear pattern necessary for this analysis.

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Most popular questions from this chapter

Peak heart rate (beats per minute) was determined both during a shuttle run and during a 300 -yard run for a sample of \(n=10\) individuals with Down syndrome ("Heart Rate Responses to Two Field Exercise Tests by Adolescents and Young Adults with Down Syndrome," Adapted Physical Activity Quarterly [1995]: 43-51), resulting in the following data: $$ \begin{array}{llllllll} \text { Shuttle } & 168 & 168 & 188 & 172 & 184 & 176 & 192 \\ \text { 300-yd } & 184 & 192 & 200 & 192 & 188 & 180 & 182 \\ \text { Shuttle } & 172 & 188 & 180 & & & & \\ \text { 300-yd } & 188 & 196 & 196 & & & & \end{array} $$ a. Construct a scatterplot of the data. What does the scatterplot suggest about the nature of the relationship between the two variables? b. With \(x=\) shuttle run peak rate and \(y=300\) -yd run peak rate, calculate \(r\). Is the value of \(r\) consistent with your answer in Part (a)? c. With \(x=300\) -yd peak rate and \(y=\) shuttle run peak rate, how does the value of \(r\) compare to what you calculated in Part (b)?

The paper "Crop Improvement for Tropical and Subtropical Australia: Designing Plants for Difficult Climates" (Field Crops Research [1991]: 113-139) gave the following data on \(x=\) crop duration (in days) for soybeans and \(y=\) crop yield (in tons per hectare): $$ \begin{array}{rrrrrr} x & 92 & 92 & 96 & 100 & 102 \\ y & 1.7 & 2.3 & 1.9 & 2.0 & 1.5 \\ x & 102 & 106 & 106 & 121 & 143 \\ y & 1.7 & 1.6 & 1.8 & 1.0 & 0.3 \end{array} $$ $$ \begin{gathered} \sum x=1060 \quad \sum y=15.8 \quad \sum x y=1601.1 \\ a=5.20683380 \quad b=-0.3421541 \end{gathered} $$ a. Construct a scatterplot of the data. Do you think the least-squares line will give accurate predictions? Explain. b. Delete the observation with the largest \(x\) value from the sample and recalculate the equation of the least-squares line. Does this observation greatly affect the equation of the line? c. What effect does the deletion in Part (b) have on the value of \(r^{2}\) ? Can you explain why this is so?

According to the article "First-Year Academic Success: A Prediction Combining Cognitive and Psychosocial Variables for Caucasian and African American Students" \((\) Journal of College Student Development \([1999]: 599\) 605), there is a mild correlation between high school GPA \((x)\) and first-year college GPA \((y)\). The data can be summarized as follows: $$ \begin{array}{clc} n=2600 & \sum x=9620 & \sum y=7436 \\ \sum x y=27,918 & \sum x^{2}=36,168 & \sum y^{2}=23,145 \end{array} $$ An alternative formula for computing the correlation coefficient that is based on raw data and is algebraically equivalent to the one given in the text is $$ r=\frac{\sum x y-\frac{\left(\sum x\right)\left(\sum y\right)}{n}}{\sqrt{\sum x^{2}-\frac{\left(\sum x\right)^{2}}{n}} \sqrt{\sum y^{2}-\frac{\left(\sum y\right)^{2}}{n}}} $$ Use this formula to compute the value of the correlation coefficient, and interpret this value.

A study was carried out to investigate the relationship between the hardness of molded plastic (y, in Brinell units) and the amount of time elapsed since termination of the molding process (x, in hours). Summary quantities include n 5 15, SSResid 5 1235.470, and SSTo 5 25,321.368. Calculate and interpret the coefficient of determination.

Is the following statement correct? Explain why or why not. A correlation coefficient of 0 implies that no relationship exists between the two variables under study.
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