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One sample problems Statistics 2126 introduction • • • • Yeah z tests are great You can do statistical tests! You simply need and Ok the mean is easy, the standard deviation, not so much • What to do, what to do… A thought… • Well let’s look at the z formula z x ( / n ) • What if we could substitute something for ? Introducing…. • The t statistic! x t (s / n ) • You do just that, put s in for Powerful technique • Think about this • You now don’t really need to know anything • A theoretical value for the population mean • And a calculated sample standard deviation The critical value of t • Unlike z, t changes depending on sample size • It has a certain number of ‘degrees of freedom’ • In our case, n-1 so if we have 22 subjects we have 21 df • We look up the critical value with 21 df An example • Four black capped chickadees in a memory experiment with two alternatives. • Their average percentage correct was 82.33 • Standard deviation was 12.5 • Are they better than chance? x t (s / n ) 82.33 50 t 12.5 /2 32.33 t 6.25 5.17 Now look up the critical value • We have 3 degrees of freedom • Let’s use an alpha of .05 (as usual) • Just use the table • Critical value is 2.35 • Our obtained value is 5.17 • Reject H0 Confidence intervals? Why not… x t(s / n ) 82.33 3.18(12.5 /2) 82.33 3.18(6.25) 82.33 3.18(6.25) 82.33 19.875 62.455 102.205 Again, just think about this • That is incredible, we can estimate, we know that accuracy of the estimate and we do not need to know ANYTHING about the population beyond some theoretical population mean • (which is usually pretty easy to figure out)