1/18/2007 219323 Probability y and Statistics for Software and Knowledge Engineers Lecture 9: Hypothesis Testing (cont.) Monchai Sopitkamon, Ph.D. Outline � Significance Levels (8.2.4) � z -Tests (8.2.5) ests (8 5) � Summary (8.3) 1
1/18/2007 Significance Levels I (8.2.4) � The null hypothesis H 0 is rejected if The null hypothesis H 0 is rejected if the p -value is smaller than size α ( significance level ), and H 0 is accepted if the p -value is larger than α . Significance Levels II (8.2.4) Error classification for hypothesis tests 2
1/18/2007 Two-Sided Hypothesis Test for a Population Mean ( ) − μ n x = 0 t s Size α tw o- sided t -test Two-Sided Hypothesis Test for a Population Mean: Example � Sample size n = 18 observations Table III provides critical points as in the left Table. Consider the test statistics | t | = α t α /2,17 3.24, 1.625, 1.74 ≤ | t | ≤ 2.898 0.10 1.740 0.05 2.110 0.01 2.898 Excel sheet 3
1/18/2007 Relationship Between Confidence Intervals & Hypothesis Tests I � The value µ 0 is contained within a 1 - α µ 0 level two-sided CI ⎛ ⎞ t s t s α − α − ⎜ − + ⎟ / 2 , n 1 / 2 , n 1 ⎜ , ⎟ x x ⎝ ⎠ n n if the p -value for the two-sided hypothesis versus H A : µ ≠ µ 0 test H 0 : µ = µ 0 0 0 A 0 is larger than α . So, if µ 0 is contained within the 1 - α level CI, the hypothesis test with size α accepts the null hypothesis, and if µ 0 is contained outside the 1 - α level CI, the hypothesis test rejects H 0 Relationship Between Confidence Intervals & Hypothesis Tests II Relationship betw een hypothesis testing and confidence intervals for tw o- sided problem s 4
1/18/2007 Relationship Between Confidence Intervals & Hypothesis Tests III: Ex.47 pg.363 � With t 0.005, 29 = 2.756 (from Table III), a 99% two-sided t -interval for the mean tensile strength is ⎛ ⎞ × × ⎛ ⎞ 2 . 756 2 . 299 2 . 756 2 . 299 t s t s ⎜ α − α − ⎟ − + = ⎜ − + ⎟ / 2 , 1 / 2 , 1 n n , 38 . 518 , 38 . 518 ⎜ ⎟ x x ⎝ ⎠ ⎝ ⎠ 30 30 n n ( ) = 37 . 36 , 39 . 67 Since µ 0 = 40.0 is not contained within this CI which is consistent with the this CI, which is consistent with the hypothesis test H 0 : µ = 40.0 versus H A : µ ≠ 40.0 having a p -value of 0.0014, so that the null hypothesis is rejected at size α = 0.01. Excel sheet Relationship Between Confidence Intervals & Hypothesis Tests III: Ex.14 pg.363 � With t 0.05, 59 = 1.671, a 90% two-sided t - interval for the mean cylinder diameter is ⎛ ⎞ ⎛ × × ⎞ t s t s 1 . 671 0 . 134 1 . 671 0 . 134 ⎜ − α − + α − ⎟ = − + ⎜ ⎟ / 2 , 1 / 2 , 1 n n ⎜ , ⎟ 49 . 999 , 49 . 999 x x ⎝ ⎠ ⎝ ⎠ 60 60 n n ( ) = 49 . 970 , 50 . 028 Since µ 0 = 50.0 is contained within this CI, which is consistent with the hypothesis test H : µ = 5 0 0 versus hypothesis test H 0 : µ = 5 0.0 versus H A : µ ≠ 50.0 having a p -value of 0.954, so that the null hypothesis is accepted at size α = 0.1. Excel sheet 5
1/18/2007 One-Sided Hypothesis Test for a Population Mean I ( ) − μ n x = 0 t s Size α one- sided t -test One-Sided Hypothesis Test for a Population Mean II ( ) − μ n x = 0 t s Size α one- sided t -test 6
1/18/2007 Relationship Between Confidence Intervals & Hypothesis Tests I Relationship betw een hypothesis testing and confidence intervals for one- sided problem s Relationship Between Confidence Intervals & Hypothesis Tests II Relationship betw een hypothesis testing and confidence intervals for one- sided problem s 7
1/18/2007 Summarization of relationships between CIs, p -values, and significance levels ( α ) for two- sided and one-sided problems One-Sided Problem Example: Ex.48 pg.366: Car Fuel Efficiency � The one-sided hypothesis are H 0 : µ ≥ 35.0 versus H A : µ < 35.0 Since the t -statistic = -1.119 > t -critical - t 0.01, 19 = -1.328, a size α = 0.10 hypothesis test accepts the null hypothesis. This is also consistent w/ the previous analysis where the p -value = 0.1386 > α = 0.10. Besides, the one-sided 90% t -interval ⎛ ⎛ ⎞ ⎞ × ⎛ ⎛ ⎞ ⎞ t s 1 . 328 2 . 915 μ ∈ ⎜ − ∞ + α − ⎟ = − ∞ + ⎜ ⎟ , 1 n ⎜ , ⎟ , 34 . 271 x ⎝ ⎠ ⎝ ⎠ 20 n ( ) = − ∞ , 35 . 14 contains the value µ 0 = 35.0, as expected. Excel sheet 8
1/18/2007 Power Levels � Significance level α � prob that null g p hypothesis is rejected when it is true (Type I error) � Small significance levels are employed in hypothesis tests so that the prob of Type I error is small. � Type II error (prob that null T pe II error (prob that n ll hypothesis is accepted when it is false) should also be minimized, thus the introduction of the “Power of a Hypothesis Test” concept. Power of a Hypothesis Test � Power = 1 – (prob of Type II error) Power 1 (prob of Type II error) = prob that the null hypothesis is rejected when it is false. � The larger the power value, the better the experiment � For a fixed significance level α , the For a fi ed significance le el the power of a hypothesis test increases as the sample size n increases. 9
1/18/2007 Outline � Significance Levels (8.2.4) � z -Tests (8.2.5) z Tests (8.2.5) � Summary (8.3) z -Tests (8.2.5) � Used when the population SD σ is Used when the population SD σ is known, rather than the sample SD s . � Uses z -statistic: ( ) − μ 0 n x = z σ which has standard normal dist when µ = µ 0 10
1/18/2007 Two-Sided z -Test I The p -value for the two-sided hypothesis testing problem testing problem versus H A : µ ≠ µ 0 H 0 : µ = µ 0 based on a data set of n observations w/ a sample mean and a population SD x σ is p -value = 2 x Φ (-|z|) where the Φ (x) is standard normal CDF and ( ) − μ 0 n x = z σ which is known as the z -statistic . Two-Sided z -Test II � A size α test rejects the null hypothesis H 0 if the test statistic | z | falls in the rejection region | z | > z α /2 and accepts the null hypothesis H 0 if the test statistic | z | falls in the acceptance region | z | ≤ z α /2 the 1 - α level two-sided CI ⎛ ⎛ σ σ ⎞ ⎞ z z μ ∈ ⎜ − α + α ⎟ / 2 / 2 , x x ⎝ ⎠ n n consists of the values µ 0 for which this hypothesis testing problem has a p -value > α , or the values µ 0 for which the size α hypothesis test accepts the null hypothesis. 11
1/18/2007 One-Sided z -Test I ( H 0 : µ ≤ µ 0 ) The p -value for the two-sided hypothesis testing problem H 0 : µ ≤ µ 0 H 0 : µ ≤ µ 0 versus versus H A : µ H A : µ > µ 0 µ 0 based on a data set of n observations w/ a sample mean and a population SD σ is x p -value = 1 – Φ (z) A size α test rejects the null hypothesis when z > z α and accepts the null hypothesis when z ≤ z α z z α σ ⎛ ⎛ ⎞ ⎞ the 1 - α level one-sided CI z μ ∈ − ∞ ⎜ α ⎟ , x ⎝ ⎠ n consists of the values µ 0 for which this hypothesis testing problem has a p -value > α , or the values µ 0 for which the size α hypothesis test accepts the null hypothesis One-Sided z -Test II ( H 0 : µ ≥ µ 0 ) The p -value for the two-sided hypothesis testing problem H 0 : µ ≥ µ 0 H 0 : µ ≥ µ 0 versus versus H A : µ H A : µ < µ 0 µ 0 based on a data set of n observations w/ a sample mean and a population SD σ is x p -value = Φ (z) A size α test rejects the null hypothesis when z < - z α and accepts the null hypothesis when z ≥ - z α z z α ⎛ ⎛ σ ⎞ ⎞ the 1 - α level one-sided CI z μ ∈ ⎜ − ∞ + α ⎟ , x ⎝ ⎠ n consists of the values µ 0 for which this hypothesis testing problem has a p -value > α , or the values µ 0 for which the size α hypothesis test accepts the null hypothesis 12
1/18/2007 Outline � Significance Levels (8.2.4) � z -Tests (8.2.5) z Tests (8.2.5) � Summary (8.3) Summary I (8.3) Decision process for inferences on a population m ean 13
1/18/2007 Summary II (8.3) Sum m ary of the t -procedure Summary III (8.3) Sum m ary of the z -procedure 14
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