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A confidence interval is an interval that covers an unknown parameter with $100(1-\alpha)\%$ confidence. Confidence intervals are a frequentist concept. They are often confused with credible intervals which is the Bayesian analog.

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Calculating Prediction Interval

Your predict.lm code is calculating confidence intervals for the fitted values. Your hand calculation is calculating prediction intervals for new data. If you want to get the same result from predict. …
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