Here, the random variable is in the middle of the interval, which is what makes the most sense in a probability statement. All is good! Robustness again: In practice we can’t find exactly for this procedure, since we a don’t know . But using the sample standard deviation to approximate s will give a good approximation. And, once we actually have a sample, we will know the value of n, so we can use a t-distribution to find a more precise value for for that sample size. a Many procedures are (that is, don’t require an approximation), but the additional complications they involve make this procedure better for explaining the basic idea. A little algebraic manipulation (which can be stated in words as, the estimate is within units of the mean then is a µ, µ “If within units of the estimate”) allows us to restate (*) as a (**) The probability that lies between - a and + a µ is approximately 0.95: P( - a < + a) is approximately0.95 µ <
Here, the random variable is in the middle of the interval, which is what makes the most sense in a probability statement. All is good! Robustness again: In practice we can’t find a exactly for this procedure, since we don’t know . But using the sample standard deviation s to approximate will give a good approximation. And, once we actually have a sample, we will know the value of n, so we can use a t-distribution to find a more precise value for a for that sample size. Many procedures are (that is, don’t require an approximation), but the additional complications they involve make this procedure better for explaining the basic idea. A little algebraic manipulation (which can be stated in words as, “If the estimate is within a units of the mean µ, then µ is within a units of the estimate”) allows us to restate (*) as (**) The probability that µ lies between - a and + a is approximately 0.95: P( - a < µ < + a) is approximately0.95
WorkshopSlidesDayTwo2017-part3.docx