Textbook Question
"In Exercises 7-12, match the description in the left column with its symbol(s) in the right column.
9. Slope
a. \(\hat{y}\)_i
b. y_i
c. b
d. (\(\bar{x}\), \(\bar{y}\))
e. m
f. \(\bar{y}\)"
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Ch. 9 - Correlation and Regression
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470
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Table of contents
- Ch. 1 - Introduction to Statistics87
- Ch. 2 - Descriptive Statistics176
- Ch. 3 - Probability184
- Ch. 4 - Discrete Probability Distributions102
- Ch. 5 - Normal Probability Distributions183
- Ch. 6 - Confidence Intervals154
- Ch. 7 - Hypothesis Testing with One Sample165
- Ch. 8 - Hypothesis Testing with Two Samples134
- Ch. 9 - Correlation and Regression87
- Ch. 10 - Chi-Square Tests and the F-Distribution87
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook
Chapter 9, Problem 9.2.12
"In Exercises 7-12, match the description in the left column with its symbol(s) in the right column.
12. The point a regression line always passes through
a. \(\hat{y}\)_i
b. y_i
c. b
d. (\(\bar{x}\), \(\bar{y}\))
e. m
f. \(\bar{y}\)"
Verified step by step guidance1
Understand that a regression line is a straight line that best fits the data points in a scatterplot, typically described by the equation or in statistics as .
Recall a key property of the least squares regression line: it always passes through the point representing the means of the x and y variables, which is .
Identify the symbols given in the problem: is the mean of x, is the mean of y, is the slope, is the y-intercept, is an observed y-value, and is a predicted y-value from the regression line.
Match the description 'The point a regression line always passes through' with the symbol representing the point of means, which is .
Conclude that the correct match is option d: .
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Regression Line
A regression line is a straight line that models the relationship between an independent variable (x) and a dependent variable (y). It is used to predict values of y based on x, minimizing the sum of squared differences between observed and predicted values.
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04:57
Using Regression Lines to Predict Values
Mean of x and y (Centroid)
The point (𝑥̄, 𝑦̄) represents the means of the x-values and y-values in the data set. The regression line always passes through this centroid, ensuring the line balances the data points around their average values.
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04:52Calculating the Mean
Regression Line Equation Components
The regression line is typically written as ŷ = b + mx, where b is the y-intercept, m is the slope, and ŷ is the predicted value of y. Understanding these symbols helps interpret the line’s position and direction.
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04:57Using Regression Lines to Predict Values
Related Practice
Textbook Question
"Predicting y-Values In Exercises 3-6, use the multiple regression equation to predict the y-values for the values of the independent variables.
6. Elephant Weight The equation used to predict the weight of an elephant (in kilograms) is
y =- 4016+11.5x_1+7.55x_2+12.5x_3
where x_1 represents the girth of the elephant (in centimeters), x_2 represents the length of the elephant (in centimeters), and x_3 represents the circumference of a footpad (in
centimeters). (Source: Field Trip Earth)
a. x_1 = 421, x_2 = 224, x_3 = 144
b. x_1 = 311, x_2 = 171, x_3 = 102
c. x_1 = 376, x_2 = 226, x_3 = 124
d. x_1 =231, x_2 = 135, x_3 = 86"
Textbook Question
"Old Vehicles In Exercises 31–34, use the figure shown at the left.
33. Coefficient of Determination Find the coefficient of determination r^2 and interpret the results."
Textbook Question
"In Exercises 19-22, two variables are given that have been shown to have correlation but no cause-and-effect relationship. Describe at least one possible reason for the correlation.
22. Marriage rate in Kentucky and number of deaths caused by falling out of a fishing boat"
Textbook Question
8. In your own words, what does it mean to say "correlation does not imply causation"? List a pair of variables that have correlation but no cause-and-effect relationship.
Textbook Question
In Exercise 25, remove the data for the international soccer player with a maximum weight of 170 kilograms and a jump height of 64 centimeters. Describe how this affects the correlation coefficient r.