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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 9.3.38
Chapter 9, Problem 9.3.38

"Confidence Intervals for y-Intercept and Slope
You can construct confidence intervals for the y-intercept B and slope M of the regression line y = Mx + B for the population by using the inequalities below.
y-intercept B :
b - E < B < b + E
where
E = t_c s_e \(\sqrt{\frac{1}{n}\) + \(\frac{\overline{x}\)^2}{\(\sum\) x^2 - \(\frac{(\Sigma x)^2}{n}\)}}
slope M :
m - E < M < m + E
where
E = \(\frac{t_c s_e}{\sqrt{\sum x^2 - \frac{(\Sigma x)^2}{n}\)}}
The values of m and b are obtained from the sample data, and the critical value t_c is found using Table 5 in Appendix B with n - 2 degrees of freedom.
In Exercises 37 and 38, construct the indicated confidence intervals for B and M using the gross domestic products and carbon dioxide emissions data found in Example 2.
38. 99% confidence interval"

Verified step by step guidance
1
Identify the sample estimates for the slope (m) and y-intercept (b) from the regression analysis of the given data.
Determine the critical t-value (t_c) corresponding to a 99% confidence level and degrees of freedom equal to n - 2, where n is the number of data points. This value can be found using a t-distribution table.
Calculate the standard error of the estimate (s_e), which measures the typical distance that the observed values fall from the regression line.
Compute the margin of error (E) for the slope using the formula: E=t_c∑x2-(∑x)2n⋅s_e and for the y-intercept using the formula: E=t_c⋅s_e⋅1n+¯x2∑x2-(∑x)2n, where ¯x is the mean of the x-values.
Construct the confidence intervals by subtracting and adding the margin of error (E) to the sample estimates: for the slope, the interval is m - E < M < m + E, and for the y-intercept, the interval is b - E < B < b + E.
Interpret the intervals as the range of plausible values for the true population slope and y-intercept with 99% confidence, meaning that if the sampling were repeated many times, approximately 99% of such intervals would contain the true parameters.

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Key Concepts

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

Confidence Interval

A confidence interval estimates a range of values within which a population parameter lies, based on sample data. It reflects the uncertainty inherent in sampling and is expressed with a confidence level, such as 99%, indicating the probability that the interval contains the true parameter.
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Introduction to Confidence Intervals

Linear Regression Parameters (Slope and Intercept)

In linear regression, the slope (M) measures the rate of change of the dependent variable with respect to the independent variable, while the y-intercept (B) is the predicted value when the independent variable is zero. Both parameters are estimated from sample data and are subject to variability.
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Guided course
05:02
Intro to Least Squares Regression Example 1

t-Distribution and Critical Value (t_c)

The t-distribution is used instead of the normal distribution when the sample size is small and the population standard deviation is unknown. The critical value t_c depends on the confidence level and degrees of freedom (n-2 here), determining the margin of error in confidence intervals for regression parameters.
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Critical Values: t-Distribution
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."

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.

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.

20. Alcohol use and tobacco use"

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"

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.

5. Black Cherry Tree Volume The volume (in cubic feet) of a black cherry tree can be modeled by the equation

y =- 52.2+0.3x_1 +4.5x_2

where x_1 is the tree's height (in feet) and x_2 is the tree's diameter (in inches). (Source: Journal of the Royal Statistical Society)

a. x_1 = 70, x_2 = 8.6

b. x_1 = 65, x_2 = 11.0

c. x_1 = 83, x_2 = 17.6

d. x_1 = 87, x_2 = 19.6"

Textbook Question

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

25. Mean Wage Construct a 99% prediction interval for the mean annual wage in Exercise 15 when the percentage of employment in STEM occupations is 13% in the industry."