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# notes2 - One-sample normal hypothesis Testing paired t-test...

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One-sample normal hypothesis Testing, paired t-test, two-sample normal inference, normal probability plots Timothy Hanson Department of Statistics, University of South Carolina Stat 704: Data Analysis I 1 / 25

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Hypothesis testing We may perform a t-test to determine whether μ is equal to some specified value μ 0 . The test statistic gives information about whether μ = μ 0 is plausible: t * = ¯ Y - μ 0 s / n . If μ = μ 0 is true, then t * t n - 1 . Rationale : Since ¯ y is our best estimate of the unknown μ , ¯ y - μ 0 will be small if μ = μ 0 . But how small is small? Standardizing the difference ¯ y - μ 0 by an estimate of sd ( ¯ Y ) = σ/ n , namely the standard error of ¯ Y , se ( ¯ Y ) = s / n gives us a known distribution for the test statistic t * before we collect data . 2 / 25
Three types of test Two sided: H 0 : μ = μ 0 versus H a : μ = μ 0 . One sided, “ < ”: H 0 : μ = μ 0 versus H a : μ < μ 0 . One sided, “ > ”: H 0 : μ = μ 0 versus H a : μ > μ 0 . If the t * we observe is highly unusual (relative to what we might see for a t n - 1 distribution), we may reject H 0 and conclude H a . Let α be the significance level of the test, the maximum allowable probability of rejecting H 0 when H 0 is true. 3 / 25

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Rejection rules Two sided: If | t * | > t n - 1 ( 1 - α/ 2 ) then reject H 0 , otherwise accept H 0 . One sided, H a : μ < μ 0 . If t * < t n - 1 ( α ) then reject H 0 , otherwise accept H 0 . One sided, H a : μ > μ 0 . If t * > t n - 1 ( 1 - α ) then reject H 0 , otherwise accept H 0 . 4 / 25
p-value approach We can also measure the evidence against H 0 using a p-value, which is the probability of observing a test statistic value as extreme or more extreme that the test statistic we did observe , if H 0 were true. A small p-value provides strong evidence against H 0 . Rule : p-value < α reject H 0 , otherwise accept H 0 . p-values are computed according to the alternative hypothesis. Let T t n - 1 ; then Two sided: p = P ( | T | > | t * | ) . One sided, H a : μ < μ 0 : p = P ( T < t * ) . One sided, H a : μ > μ 0 : p = P ( T > t * ) . 5 / 25

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Example We wish to test whether the true mean high temperature is greater than 75 o using α = 0 . 01: H 0 : μ = 75 versus H a : μ > 75 . t * = 77 . 667 - 75 8 . 872 / 30 = 1 . 646 < t 29 ( 0 . 99 ) = 2 . 462 . What do we conclude? Note that p = 0 . 05525 > 0 . 01. 6 / 25
Connection between CI and two-sided test An α -level two-sided test rejects H 0 : μ = μ 0 if and only if μ 0 falls outside the ( 1 - α ) 100 % CI about μ .

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notes2 - One-sample normal hypothesis Testing paired t-test...

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