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# IE330_Chapter8 - Chapter 8 Statistical Intervals for a...

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You should be able to recall: How to standardize a normally distributed random variable How to construct a t distribution from a normally distributed random variable The formula for lower- and upper-confidence limits The difference between 99% and 95% confidence intervals The difference between one- and two-sided confidence bounds You should be able to accomplish: Construct confidence intervals on the mean of a normal distribution Construct confidence intervals on the variance and standard deviation of a normal distribution Construct confidence intervals on a population proportion Construct prediction intervals for a future observation Construct a tolerance interval for a normal population Determine the sample size which will bound the error on an estimate You should be able to understand: The difference between confidence, tolerance, and prediction intervals What a standard normal distribution is. When a t distribution is used. When a chi-squared distribution is used. Chapter 8: Statistical Intervals for a Single Sample

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In the last set of lectures, we discussed how we can estimate population parameters from a set of samples. Consider a company that makes 5 mm diameter rivets. We’re interested in the average diameter of the rivets and the variance of that diameter. We take a random sample of 5,000 rivets. We measure the diameter of each and record it. Recall that if the samples are independent, and come from the same distribution, we can use the mean and variance of the sample to estimate the mean and variance of the population. Let’s go ahead and do that. To do so, I’m going to generate a set of random data from a known population. Normally we wouldn’t know the population parameters of course, but in order to generate the data Excel (or Minitab or whatever statistical package you use) needs to know the distribution. Confidence intervals: example
As mentioned in the last set of slides, we as yet don’t have any information on how good these estimates are. What we need are “confidence intervals” on the mean and variance estimates, where we are 90% or 95%, or 99% confident that the true mean lies within some range. Note that this statement is not technically accurate . Really this will mean that 90% or 95% or 99% of the confidence intervals we (could) create will actually include µ. This is not to be confused with two other terms: Tolerance interval: this is where 90%, or 95%, or 99% of all values lie within some range. This has nothing to do with estimates of mean or variance. Prediction interval: this is similar to a tolerance interval, except that this interval bounds the values of future samples. This also has nothing to do with estimates of mean or variance.

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