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Ch. 9 - Correlation and Regression
Larson - Elementary Statistics: Picturing the World 8th Edition
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoLarson 8th EditionCh. 9 - Correlation and RegressionProblema 9.3.38
Capitolo 9, Problema 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"

Guida verificata passo dopo passo
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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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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06:33
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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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
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