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STA291 Statistical Methods Lecture 29

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SE, and the confidence interval, becomes smaller with increasing n. SE, and the confidence interval, are larger for samples with more spread around the line (when s e is larger). Standard Errors for Mean Values Confidence Interval for the Mean Response Last time, we said we were modeling our line to infer the “line of means”—the expected value of our response variable for each given value of the explanatory variable. The confidence interval for the mean response, v, at a value x v, is: where:

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Standard Errors for Mean Values Confidence Interval for the Mean Response The confidence interval for the mean response, v, at a value x v, is: where: SE becomes larger the further x ν gets from. That is, the confidence interval broadens as you move away from. (See figure at right.)

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Standard Errors for Predicted Values Because of the extra term, the confidence interval for individual values is broader that those for the predicted mean value. Prediction Interval for an Individual Response Now, we tackle the more difficult (as far as additional variability) of predicting a single value at a value x v. When conditions are met, that interval is: where:

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Difference Between Confidence and Prediction Intervals Confidence interval for a mean: The result at 95% means: “We are 95% confident that the mean value of y is between 4.40 and 4.70 when x = 10.1.”

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Prediction interval for an individual value: The result at 95% means: “We are 95% confident that a single measurement of y will be between 3.95 and 5.15 when x = 10.1.” Difference Between Confidence and Prediction Intervals

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Using Confidence and Prediction Intervals Example : External Hard Disks A study of external disk drives reveals a linear relationship between the Capacity (in GB) and the Price (in $). Regression resulted in the following: Price = 18.64 + 0.104Capacity s e = 17.95, and SE(b 1 ) = 0.0051 Find the predicted Price of a 1000 GB hard drive. Find the 95% confidence interval for the mean Price of all 1000 GB hard drives. Find the 95% prediction interval for the Price of one 1000 GB hard drive.

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Example : External Hard Disks A study of external disk drives reveals a linear relationship between the Capacity (in GB) and the Price (in $). Regression resulted in the following: Price = 18.64 + 0.104Capacity s e = 17.95, and SE(b 1 ) = 0.0051 Find the predicted Price of a 1000 GB hard drive. Using Confidence and Prediction Intervals Price = 18.64 + 0.104(1000) = 122.64

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Example : External Hard Disks A study of external disk drives reveals a linear relationship between the Capacity (in GB) and the Price (in $). Regression resulted in the following: Price = 18.64 + 0.104Capacity s e = 17.95, and SE(b 1 ) = 0.0051 Find the 95% confidence interval for the mean Price of all 1000 GB hard drives. Using Confidence and Prediction Intervals

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Example : External Hard Disks A study of external disk drives reveals a linear relationship between the Capacity (in GB) and the Price (in $). Regression resulted in the following: Price = 18.64 + 0.104Capacity s e = 17.95, and SE(b 1 ) = 0.0051 Find the 95% prediction interval for the price of one 1000 GB hard drive. Using Confidence and Prediction Intervals

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Looking back o Construct and interpret a confidence interval for the mean value o Construct and interpret a prediction interval for an individual value

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