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Ch. 9 - Correlation and Regression
Larson - Elementary Statistics: Picturing the World 8th Edition
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook
All textbooksLarson 8th EditionCh. 9 - Correlation and RegressionProblem 9.Q.8
Chapter 9, Problem 9.Q.8

"[APPLET] For Exercises 1–8, use the data in the table, which shows the average annual salaries (both in thousands of dollars) for secondary and elementary school teachers, excluding special and vocational education teachers, in the United States for 11 years. (Source: U.S. Bureau of Labor Statistics)
Table displaying average annual salaries in thousands for secondary and elementary school teachers over 11 years.
8. Construct a 95% prediction interval for the average annual salary of elementary school teachers when the average annual salary of secondary school teachers is \$63,500. Interpret the results."

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Step 1: Identify the variables and the goal. Here, x represents the average annual salary of secondary school teachers, and y represents the average annual salary of elementary school teachers. We want to construct a 95% prediction interval for y when x = 63.5 (in thousands of dollars).
Step 2: Perform a linear regression analysis to find the regression equation \( \hat{y} = b_0 + b_1 x \). Calculate the slope \( b_1 \) and intercept \( b_0 \) using the formulas: \[ b_1 = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2} \quad \text{and} \quad b_0 = \bar{y} - b_1 \bar{x} \] where \( \bar{x} \) and \( \bar{y} \) are the means of x and y respectively.
Step 3: Calculate the predicted value \( \hat{y} \) for \( x = 63.5 \) by substituting into the regression equation: \[ \hat{y} = b_0 + b_1 (63.5) \].
Step 4: Compute the standard error of the prediction, which accounts for both the error in estimating the mean response and the variability of individual observations. The formula for the standard error of prediction is: \[ SE_{pred} = s \sqrt{1 + \frac{1}{n} + \frac{(x_0 - \bar{x})^2}{\sum (x_i - \bar{x})^2}} \] where \( s \) is the standard error of the estimate, \( n \) is the number of data points, and \( x_0 = 63.5 \).
Step 5: Determine the critical t-value \( t^* \) for a 95% confidence level with \( n-2 \) degrees of freedom. Then construct the 95% prediction interval using: \[ \hat{y} \pm t^* \times SE_{pred} \]. Finally, interpret this interval as the range in which we expect the salary of an individual elementary school teacher to fall when the secondary school teacher salary is \$63,500.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Prediction Interval

A prediction interval estimates the range within which a single new observation is expected to fall, with a certain level of confidence (e.g., 95%). It accounts for both the uncertainty in estimating the regression line and the variability of individual data points around that line.
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Linear Regression

Linear regression models the relationship between two variables by fitting a straight line that minimizes the sum of squared differences between observed and predicted values. It allows prediction of the dependent variable (elementary salaries) based on the independent variable (secondary salaries).
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Intro to Least Squares Regression

Interpretation of Confidence and Prediction Intervals

While confidence intervals estimate the mean response for a given predictor value, prediction intervals provide a wider range to predict individual outcomes. Interpreting a 95% prediction interval means we are 95% confident the actual salary will fall within this range for a given secondary teacher salary.
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Related Practice
Textbook Question

"Constructing and Interpreting a Prediction Interval In Exercises 21-30, construct the indicated prediction interval and interpret the results.

26. Voter Turnout Construct a 99% prediction interval for number of ballots cast in Exercise 16 when the voting age population is 210 million."

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Textbook Question

In Exercise 26, add data for an international soccer player who can perform the half squat with a maximum of 210 kilograms and can sprint 10 meters in 2.00 seconds. Describe how this affects the correlation coefficient r.

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Textbook Question

"9. Stock Price The equation used to predict the stock price (in dollars) at the end of the year for a restaurant chain is y=- 86+7.46x_1 - 1.61x_2

where x_1 is the total revenue (in billions of dollars) and x_2 is the shareholders' equity (in

billions of dollars). Use the multiple regression equation to predict the y-values for the

values of the independent variables.

a. x_1 = 27.6, x_2 = 15.3

b. x_1 = 24.1, x_2 = 14.6

c. x_1 = 23.5, x_2 = 13.4

d. x_1 = 22.8, x_2 =15.3"

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Textbook Question

"In Exercises 7-10, use the value of the correlation coefficient r to calculate the coefficient of determination r^2. What does this tell you about the explained variation of the data about the regression line? about the unexplained variation?

10. r =0.881"

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Textbook Question

"In Exercises 19-24, construct the indicated prediction interval and interpret the results.

24. Construct a 99% prediction interval for the price of a gas grill in Exercise 18 with a usable cooking area of 900 square inches."

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Textbook Question

"In Exercises 17 and 18, use the data to (a) find the coefficient of determination r^2 and interpret

the result, and (b) find the standard error of estimate s_e and interpret the result.


18. [APPLET] The table shows the cooking areas (in square inches) of 18 gas grills and their prices (in dollars). The regression equation is y = 1.501x - 341.501. (Source: Lowe's)

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