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## Summary of t-Test for Testing a Single Population Mean ( m )

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**t-Test Statistic**• Assumptions: • Population is normal although this assumption can be relaxed if sample size is “large”. • Random sample was drawn from the population of interest.**¥**d f = [ i . e . , N o r m a l ( 0 , 1 ) ] d f = 5 d f = 2 4 2 0 2 4 - - The t-distribution Student(df ) density curves for various df. 1. The t-distribution has one parameter that controls it’s shape called thedegrees of freedom.**¥**d f = [ i . e . , N o r m a l ( 0 , 1 ) ] d f = 5 d f = 2 4 2 0 2 4 - - The t-distribution 2. The Student’s t-distribution is bell shaped and centred at 0 — like the Standard Normal distribution but more variable (larger spread).**¥**d f = [ i . e . , N o r m a l ( 0 , 1 ) ] d f = 5 d f = 2 4 2 0 2 4 - - The t-distribution 3. As df increases, the t-distribution becomes more and more like the standard normal.**¥**d f = [ i . e . , N o r m a l ( 0 , 1 ) ] d f = 5 d f = 2 4 2 0 2 4 - - The t-distribution 4. t-dist (df = ¥) and Normal(0, 1) are two ways of describing the same distribution.**The t-distribution**From now on we will treat ashaving at-distribution (df = n - 1). For confidence intervals we will build t-standard-error intervals, estimate ± (t-quantile value) SE(estimate)**The t-distribution**Example: P(-1.96 £ Z£ 1.96) = 0.95 (standard normal) P(-2.365 £ t£ 2.365) = 0.95 for t-dist. w/ df = 7 Hence, if we are taking samples of size n = 8 and we want to build intervals that include m for 95% of all samples taken in the long run, then we use**P-value**0 t P-value 0 t t is negative Form of Hypotheses Ho: m = mo HA: m < mo (lower-tail test) Ho: m = mo HA: m > mo (upper-tail test) t is positive P-values are computed by finding areas beneath a t-distribution (df = n – 1)**0**-t t Form of Hypotheses Ho: m = mo HA: m mo (two-tailed test) P-value = Shaded Area t is either pos. or neg. This test is equivalent to constructing a 100(1-a)% CI for m and checking in mois contained in the resulting interval. Reject Ho if the CI does not cover mo.**t-Probability Calculator in JMP**Enter test statistic value ( t ) and df in these cells and the tail probabilities will update automatically.**t-Quantile Calculator for CI’s**The t-table value or standard error multiplier for the desired confidence level appears here. Enter desired confidence level which is typically 90, 95, or 99.