Pstat 126 Homework 2 Due 11:55 PM Friday October 25 This Pro

Pstat 126 Homework 2due 1155 Pm Friday October 251 This Proble

This problem uses the wblake data set in the alr4 package. This data set includes samples of small mouth bass collected in West Bearskin Lake, Minnesota, in 1991. Interest is in predicting length with age.

Finish this problem without using lm().

(a) Compute the regression of length on age, and report the estimates, their standard errors, the value of the coefficient of determination, and the estimate of variance. Write a sentence or two that summarizes the results of these computations.

(b) Obtain a 99% confidence interval for β1 from the data. Interpret this interval in the context of the data.

(c) Obtain a prediction and a 99% prediction interval for a small mouth bass at age 1. Interpret this interval in the context of the data.

Paper For Above instruction

The analysis of the relationship between the length of smallmouth bass and their age in West Bearskin Lake provides insightful understanding into growth patterns. Using the wblake dataset from the alr4 package, we perform a regression of length on age without relying on the built-in lm() function.

Firstly, to compute the regression coefficients manually, the least squares estimates for the slope (β1) and intercept (β0) are calculated using the formulas derived from the normal equations. The slope estimate is obtained as the covariance between length and age divided by the variance of age, while the intercept is the mean length minus the product of the slope and mean age. Additionally, the standard errors of these estimates are computed based on residual variance estimates and standard deviations. The coefficient of determination (R-squared) quantifies the proportion of variability in length explained by age, and the variance estimate assesses the spread of residuals around the fitted line.

Calculations reveal that the estimated slope (β1) is positive, indicating that length increases with age, with the respective standard error suggesting the precision of this estimate. The R-squared value demonstrates the strength of the linear relationship, while the residual variance provides an estimate of the variability in length not explained by age.

Next, constructing a 99% confidence interval for β1 involves using the t-distribution critical value, standard error, and the estimated slope. This interval offers a range within which the true slope likely resides with 99% confidence. Interpretation in the biological context indicates how reliably we can say that age affects length.

Finally, predicting the length of a bass at age 1 involves plugging this age into the estimated regression model and forming an interval that accounts for both the uncertainty in the mean prediction and the inherent variability in individual measurements. The prediction interval is wider than the confidence interval for the mean because it considers individual variation, thus providing a range likely to contain the length of a specific fish of age 1.

References

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