Lecture 13_one sample t-test

Lecture 13_one sample t-test - 10/18/2010 One Sample t-test...

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10/18/2010 1 One Sample t-test 1 Hypothesis Testing with z-tests • Z-tests provided an easy way to illustrate the logic of one-sample hypothesis tests, but it was necessary to know both the mean (μ) and standard deviation (σ) of the population to use the test… in most cases this is not realistic • When the value of σ is not known, we will estimate σ from the population and use a different statistical test called the the t-test 2 Overview of the t-test • The t-test is almost identical to the z-test, but the sample standard deviation (s) is used in the formula rather than a known population standard deviation (σ) • Like the z-test, it converts a sample mean into a test statistic (here a t-score instead of a z-score) and compares it to a critical value on a distribution associated with the null hypothesis to determine the probability value associated with the statistic 3
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10/18/2010 2 History of the t-statistic • A statistician, William Sealy Gossett, worked as a chemist at the Guinness Brewing Company doing quality control studies for brewing beer using small samples of barley • After looking at his results over numerous experiments, he noticed that the means did not follow always normal curve, as predicted by the central limit theorem 4 History of the t-statistic • Instead of following a normal curve, the sampling distributions for smaller samples of data (such as the barley varieties Gossett studied at the brewery) were somewhat flatter than the normal curve and the shape of the distributions depended on the size of the sample – These sampling distributions were t- distributions 5 History of the t-statistic • Gossett wanted to publish his findings, but due to contractual agreements with Guinness he was not allowed – A researcher at Guinness had previously published a paper exposing trade secrets, so Guinness prohibited its employees from publishing any papers regardless of content – So Gossett published his results under a fake name, “Student”, and called his statistic Student’s t statistic 6
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10/18/2010 3 William Sealy Gossett (“Student”) 7 More Details On The t-test • The population standard deviation need not be known to compute the statistic • The sample standard deviation is used as a replacement in the formula – Recall: sample standard deviation is the best estimator of the population standard deviation • Like a distribution of z-scores, t-values have a mean of 0 and standard deviation of 1 8 Compare z-test and t-test • The z-score formula was: • Substituting the sample standard deviation into the formula gives a t- statistic: n σ μ M z X n s μ M t X X M σ X M s 9
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10/18/2010 4 The t Sampling Distribution • There is not a single sampling distribution for the t statistic, but a family of distributions (like the normal curve) • Each unique sample size will yield a slightly different sampling distribution • This needs to be taken into account when computing the probability value associated
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This note was uploaded on 02/17/2011 for the course PYSC 227 taught by Professor Fairchild during the Spring '10 term at South Carolina.

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Lecture 13_one sample t-test - 10/18/2010 One Sample t-test...

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