Student s t Distribution Lecture 36 Section 10 2 Fri Nov 11 2005 What if is Unknown It is more realistic to assume that the value of is unknown If we don t know the value of then we probably don t know the value of In this case we use s to estimate What if is Unknown Let us assume that the population is normal or nearly normal Then the distribution of x is normal That is x n is N 0 1 However for small n x is not N 0 1 s n What if is Unknown If it is not N 0 1 then what is it Student s t Distribution It has a distribution called Student s t distribution x t s n The t distribution was discovered by W S Gosset in 1908 See http mathworld wolfram com Studentst Dis tribution html The t Distribution The shape of the t distribution is very similar to the shape of the standard normal distribution It is symmetric unimodal centered at 0 But it is wider than the standard normal That is because of the additional variability introduced by using s instead of The t Distribution Furthermore the t distribution has a slightly different shape for each possible sample size As n gets larger and larger s exhibits less and less variability so the shape of the t distribution approaches the standard normal In fact if n 30 then the t distribution is approximately standard normal Degrees of Freedom If the sample size is n then t is said to have n 1 degrees of freedom We use df to denote degrees of freedom We will use the notation t df to represent the t distribution with df degrees of freedom For example t 5 is the t distribution with 5 degrees of freedom i e sample size 6 Standard Normal vs t Distribution The distributions t 2 t 30 and N 0 1 t 2 t 30 N 0 1 Decision Tree Is known yes Is the population normal yes no X Z n TBA Is n 30 yes X Z n no no Give up Decision Tree Is known yes Is the population normal yes no X Z n Is the population normal yes no Is n 30 yes X Z n no no Give up Decision Tree Is known yes Is the population normal yes no X Z n Is the population normal yes no Is n 30 Is n 30 yes X Z n no no Give up yes no Decision Tree Is known yes Is the population normal yes no X Z n Is the population normal yes no Is n 30 Is n 30 yes X Z n no no yes Give t n 1 X s n up t n 1 Z no Decision Tree Is known yes Is the population normal yes no X Z n Is the population normal yes no Is n 30 Is n 30 yes X Z n no no yes no Give t n 1 X t n 1 X s n s n up t n 1 Z Decision Tree Is known yes Is the population normal yes no X Z n Is the population normal yes no Is n 30 Is n 30 yes X Z n no no yes no Give t n 1 X t n 1 X s n s n up t n 1 Z Is n 30 yes no Decision Tree Is known yes Is the population normal yes no X Z n Is the population normal yes no Is n 30 Is n 30 yes X Z n no no yes no Give t n 1 X t n 1 X s n s n up t n 1 Z Is n 30 yes Z X s n no Decision Tree Is known yes Is the population normal yes no X Z n Is the population normal yes no Is n 30 Is n 30 yes X Z n no no yes no Give t n 1 X t n 1 X s n s n up t n 1 Z Is n 30 yes Z X s n no Give up Decision Tree Is known yes Is the population normal yes no X Z n Is the population normal yes no Is n 30 Is n 30 yes X Z n no no yes no Give t n 1 X t n 1 X s n s n up t n 1 Z Is n 30 yes Z X s n no Give up Decision Tree Is known yes Is the population normal yes no X Z n Is the population normal yes no Is n 30 Is n 30 yes X Z n no no yes no Give t n 1 X t n 1 X s n s n up t n 1 Z Is n 30 yes Z X s n no Give up Table IV t Percentiles Table IV gives certain percentiles of t for certain degrees of freedom Specific percentiles for upper tail areas 0 40 0 30 0 20 0 10 0 05 0 025 0 01 0 005 Specific degrees of freedom 1 2 3 30 40 60 120 Table IV t Percentiles The table tells us for example that P t 1 812 0 05 when df 10 Since the t distribution is symmetric we can also use the table for lower tails by making the t values negative So what is P t 1 812 when df 10 Table IV t Percentiles The table allows us to look up certain percentiles but it will not allow us to find probabilities in general TI 83 Student s t Distribution The TI 83 will find probabilities for the t distribution but not percentiles in general Press DISTR Select tcdf and press ENTER tcdf appears in the display Enter the lower endpoint Enter the upper endpoint TI 83 Student s t Distribution Enter the number of degrees of freedom n 1 Press ENTER The result is the probability Example Enter tcdf 1 812 E99 10 Enter tcdf E99 1 812 10 The result is 0 0500 The result is 0 0500 Thus P t 1 812 0 05 when there are 10 degrees of freedom n 11 Hypothesis Testing with t We should use the t distribution if The population is normal or nearly normal and is unknown so we use s in its place and The sample size is small n 30 Otherwise we should not use t Either use Z or give up Hypothesis Testing with t The hypothesis testing procedure is the same except for two steps Step 3 Find the value of the test statistic The test statistic is now x 0 t s n Step 4 Find the p value We must look it up in the t table or use tcdf on the TI 83 Example Re do Example 10 1 p 616 by hand under the assumption that is unknown TI 83 Hypothesis Testing When is Unknown Press STAT Select TESTS Select T Test A window appears requesting information Choose Data or …
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