Why bother with t ? If you don’t know the population standard deviation, σ, but you still want to use a sample to find a confidence interval. t builds in a little more uncertainty based on the lack of a trustworthy σ. The plan: 1.This lesson – learn about t and areas and critical values, much like we have done with z. 2.Next lesson – doing confidence intervals with t.
Why bother with t ? Observe in the picture how t isn’t quite as high and bold in the middle part of the bell curve. The uncertainty shows up as extra area in the tails of the bell curve. As the sample size n gets larger, the degrees of freedom d.f. gets larger, and the uncertainty becomes less uncertain, and the t bell curve gets very much closer to the normal distribution bell curve we use in z problems. History of t : Q.A. at an Irish brewery circa 1900. See textbook or internet for all the details.
Example 8.11: Finding the Value of t Given the Area to the Left, with Excel Excel: T.INV(area to the left of t, df), same thing. Excel special if you know area in two tails total: =T.INV.2T(area in two tails total, df)
Example 8.12: Finding the Value of t Given the Area in Two Tails (cont.) – with Excel Recall: we seek t and –t such that two tails total area 0.02, d.f. = 7 Excel with convenient =T.INV.2T(total area, d.f.) Or Excel with one-tailed version, manually divide area by 2: = T.INV(one tailed area, d.f.)