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STAT 2 – INFERENCE BASED ON THE t DISTRIBUTION The previous inference procedures based on the Z distribution assume that the population standard deviation σ is known. But if σ is not known, the t distribution must be used. Brief History : The t distribution is due to W.S. Gosset, who discovered it while working as a chemist for the Guinness Brewery in Dublin. He used statistics to improve the yield of barley used to make beer and to solve small-sample problems that arose in the quality control of beer production. Because of the brewery’s policy of keeping their research findings secret, Gosset had to publish his findings under the pen name Student; today, the t distribution is also known as Student’s t . (Gosset published his discovery in 1908 and died in 1937, two years after he became head brewer for the Guinness Brewery in London.) Properties of the t Distribution : The distribution has one parameter, called the degrees of freedom ( df ), that determines the variance and spread of the distribution. Like the Z distribution, its density curve is symmetric and bell-shaped, has total area equal to 1, and is centered at the mean 0. However, its SD σ is greater than 1 (recall that Z has an SD of 1); so the t curve has more spread than the normal curve. Furthermore, the SD depends on the df in such a way that the greater the df , the smaller is the SD and the spread of the distribution. As the df goes to infinity, the SD of the t distribution approaches 1 and, in fact, the t distribution approaches the standard normal distribution in the limit. Various upper tail quantiles (critical values) of the t distribution for various values of the df are given in the t table, Table C in the appendix of the textbook. The top row of the table contains a range of upper tail areas, while the left margin of the table contains a range of df’s . Found at the intersection of a particular df and a particular upper tail area is the * t quantile for that df and tail area. The last row of the t table is useful because it contains the * t values when the df
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This note was uploaded on 07/08/2008 for the course STAT 1 taught by Professor Sugahara during the Spring '08 term at University of California, Berkeley.

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