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10_1_sample_t - Thettest Inferences about Population Means...

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    The t-test Inferences about Population Means when population SD is unknown
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    Confidence intervals in z (Review) Want to estimate height of students at USF. Sampled N=100 students. Found mean =68 in and SD = 6 in. Best guess for population mean is 68 inches plus or minus some. 95%CI = 95%CI=68±(1.96)[6/sqrt(100)] 68 ±1.96(.6) = 68 ±1.18 Interval is 66.82 to 69.18. Such an interval will contain the mean 95% of the time. X z X σ 05 . ± N X X σ σ =
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    Problem with z Formulas so far use population SD, and they have been correct, but SD is usually unknown, so we have to estimate Estimate will be off a bit; would be nice to account for this The statistic called ‘ t ’ adjusts for error in estimate of SD. Estimate of SD is better as sample size increases, so t changes with N. The values of t are basically the same as z , but t spreads out more and more as the sample size gets small.
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    The  t  Distribution We use t when the population variance is unknown (the usual case) and sample size is small (N<100, the usual case). If you use a stat package for testing hypotheses about means, you will use t . The t distribution is a short, fat relative of the normal. The shape of t depends on its df. As N becomes infinitely large, t becomes normal.
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    Example values from  t  and  z Area
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