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Statistics 270– Lecture 25 Cautions about Z-Tests • • • • Data must be a random sample Outliers can distort results Shape of the population distribution matters (large and small samples?) For significance tests, there is a difference between practical and statistical significance Types of Errors in a Significance Test • Suppose that the null hypothesis is true, we could collect a sample that suggests that we reject H0 • Suppose that H0 is not true, we could fail to reject the null hypothesis Inference About the Population Mean • To make inference about the population mean, m, we have used the z-test • Key feature: s is known • Most often, s is unknown! • • s must be estimated from the data Use to estimate s Inference About the Population Mean • To make probability statements about the the sample mean, we have used the Z-statistic when s is known • When s is unknown, we use the one-sample t-statistic with (n-1) degrees of freedom Inference About the Population Mean • Standard error: • Degrees of freedom: One Sample t-Test for a Population Mean • • • Data: random sample x1, x2, …, xn Mean, m, is unknown Standard deviation is unknown • For testing the hypothesis H0: m=m0 • Test Statistic: • t x m0 s/ n Degrees of freedom: t has a t-distribution with n-1 degrees of freedom One Sample t-Test for a Population Mean • Computing p-value depends on the alternate hypothesis: Alternate hypothesis H1: m m 0 H1: m m 0 H1: m m 0 • P-Value P (T t ) P(T t ) 2 P(T t ) P-values are exact if the population distribution is normal and approximately correct for large samples in other cases One Sample t-Test for a Population Mean Rule of Thumb • For n<15, use t-test if data appear approximately normal • For n 15, can use t-test when no outliers • For large n (say n 40), can safely use t-test • Note: always require random sample! Example • Composition of earth’s atmosphere has changed over time • Gas bubbles in ancient amber are examined to study the nature of the atmosphere long ago • Measurements on specimens of amber from the Cretaceous period (75-95 million years ago) give the following percentages of nitrogen 63.4 65.0 64.4 63.3 54.8 64.5 60.8 49.1 51.0 Example • Assume the data are a random sample from the Cretaceous period • To see if there is a difference with today’s 78.1% nitrogen, conduct a hypothesis test using these data Column 1 70 65 60 55 50 45 Example • Hypotheses: • Test Statistic • P-value • Conclusion