Lec18 - 7-3 Prediction Interval (PI) Single Future Value...

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Unformatted text preview: 7-3 Prediction Interval (PI) Single Future Value Future Value IOE/Stat 265 Fall 2009 IOE/Stat 265, Fall 2009 Lecture Lecture #18: Prediction and Tolerance Intervals Fi conside the best point estimate of single First consider the best point estimate of a single future value from a normal distribution. Predict X n+1 with x with This estimate is unbiased Ch 7.3 E(X n+1 - x) = E(X n+1 ) - E(x) = μ − μ = 0 The variance of the point estimate error V(X n+1 - x) = V(X n+1 ) + V(x) = σ2 + 1 1 σ2 1⎞ ⎛ = σ2 ⎜ 1 + ⎟ n n⎠ ⎝ 2 7-3 Prediction Interval (PI) Single Future Value Future Value If we don’t know If we don’t know σ2 then estimate with s2 estimate with and the interval estimate for a future observation is: observation is: Example 1: Bottle Filling Machine 1: Bottle Filling Machine Suppose fill volume has historically followed Suppose fill volume has historically followed a normal distribution with mean, μ = 10 and standard deviation standard deviation, σ = 2. You take a sample You take sample of n=9 bottles. Provide a 95% prediction interval for the next (e 10 bottle interval for the next (e.g. 10th) bottle. X ± t α /2,n−1 s 1+ 1 n 3 4 Example (continued) Example (continued) Suppose you estimated the standard Suppose you estimated the standard deviation, σ, with s = 2. Provide a 95% prediction interval for the 10 prediction interval for the 10th observation: 7-3 Tolerance Interval for a Normal Distribution Normal Distribution Suppose you were asked to provide an interval Suppose you were asked to provide an interval estimate for a certain proportion of future observations. For example, you might predict that 95% of future observations would lie in the interval observations would lie in the interval x − 1.96 s , x + 1.96 s How often would 95% of future observations lie in this interval? 6 8 7-3 Tolerance Interval for a Normal Distribution Normal Distribution Suppose you were asked to provide an interval Suppose you were asked to provide an interval estimate for a certain proportion of future observations. For example, you might predict that 95% of future observations would lie in the interval observations would lie in the interval Tolerance Interval Interval A tolerance interval for capturing at least γ % of the tolerance interval for capturing at least of the values in a normal distribution with a confidence 100(1 – α)% is (x − k s , x + k s) x − 1.96 s , x + 1.96 s How often would 95% of future observations lie in this interval? 9 Where k is a “tolerance factor” found in Appx A-6. Values provided for γ = 90%, 95%, 99% 90% 95% 99% (1 – α)% = 90%, 95%, 99% 10 Example: Bottle Filling (continued) Example: Bottle Filling (continued) Suppose you found sample Suppose you found sample x = 10 and s = 2 10 and based on a random sample of n=9 bottles. Provide 95% tolerance interval for 95% of Provide a 95% tolerance interval for 95% of future observations : Suppose (1 Suppose (1 - α) % = 99% what will be the 99% what will be the interval width? Suppose we double the number of samples to do the of samples to n = 18. What impact would this have on the above interval? above interval? 11 11 15 Tolerance Calculator.xls Calculator N= 1-α = K= X-Bar S Upper Limit Lower Limit γ 22 90% 95% 2.2637 10 2 14.5274 5.4726 700 90% 50% 1.6468 10 2 13.2936 6.7064 700 90% 95% 1.7220 10 2 13.4441 6.5559 http://www.itl.nist.gov/div898/handbook/prc/section2/prc263.htm http://www.nist.gov/ National Institute of Standards and Technology Keyword Search: tolerance intervals 18 ...
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