χ 2 test Recap p -value Two sample tests Other goodness-of-fit checks Dataanalyse - Hypotesetest - Kursusgang 3 Ege Rubak - rubak@math.aau.dk http://www.math.aau.dk/ ∼ rubak/teaching/2010/nano4 19. februar 2010 1/20 Dataanalyse - Kursusgang 3
χ 2 test Recap p -value Two sample tests Other goodness-of-fit checks Confidence interval for µ with known σ Sample with ( X 1 , . . . , X n ) independent, X i ∼ N ( µ, σ 2 ) . � We have: X ∼ N ( µ, σ 2 / n ) and Z = X − µ � ∼ N ( 0 , 1 ) . σ 2 / n � Therefore: � � z α/ 2 ≤ Z ≤ z 1 − α/ 2 = 1 − α, P where z α/ 2 is the α/ 2 quantile in N ( 0 , 1 ) . I.e. if α is 0 . 05 as usual we look up the 2 . 5 % quantile z 0 . 025 = − 1 . 96 and the 97 . 5 % quantile z 0 . 975 = 1 . 96 in the standard normal distribution N ( 0 , 1 ) . � I.e. the confidence interval is: σ σ √ n ; x + z 1 − α/ 2 √ n ] [ x + z α/ 2 2/20 Dataanalyse - Kursusgang 3
χ 2 test Recap p -value Two sample tests Other goodness-of-fit checks Confidence interval for µ with known σ Sample with ( X 1 , . . . , X n ) independent, X i ∼ N ( µ, σ 2 ) . � We have: X ∼ N ( µ, σ 2 / n ) and Z = X − µ � ∼ N ( 0 , 1 ) . σ 2 / n � Therefore: � � z α/ 2 ≤ Z ≤ z 1 − α/ 2 = 1 − α, P where z α/ 2 is the α/ 2 quantile in N ( 0 , 1 ) . I.e. if α is 0 . 05 as usual we look up the 2 . 5 % quantile z 0 . 025 = − 1 . 96 and the 97 . 5 % quantile z 0 . 975 = 1 . 96 in the standard normal distribution N ( 0 , 1 ) . � I.e. the confidence interval is: σ σ √ n ; x + z 1 − α/ 2 √ n ] [ x + z α/ 2 2/20 Dataanalyse - Kursusgang 3
χ 2 test Recap p -value Two sample tests Other goodness-of-fit checks Confidence interval for µ with known σ Sample with ( X 1 , . . . , X n ) independent, X i ∼ N ( µ, σ 2 ) . � We have: X ∼ N ( µ, σ 2 / n ) and Z = X − µ � ∼ N ( 0 , 1 ) . σ 2 / n � Therefore: � � z α/ 2 ≤ Z ≤ z 1 − α/ 2 = 1 − α, P where z α/ 2 is the α/ 2 quantile in N ( 0 , 1 ) . I.e. if α is 0 . 05 as usual we look up the 2 . 5 % quantile z 0 . 025 = − 1 . 96 and the 97 . 5 % quantile z 0 . 975 = 1 . 96 in the standard normal distribution N ( 0 , 1 ) . � I.e. the confidence interval is: σ σ √ n ; x + z 1 − α/ 2 √ n ] [ x + z α/ 2 2/20 Dataanalyse - Kursusgang 3
χ 2 test Recap p -value Two sample tests Other goodness-of-fit checks Hypothesis test for µ = µ 0 with known σ � Suppose we have a hypothesis H 0 : µ = µ 0 ( µ 0 is a known number). Assuming H 0 is true we can calculate Z = X − µ 0 � ∼ N ( 0 , 1 ) σ 2 / n for a given data set. This is called the Z -statistic, and if Z / ∈ [ z α/ 2 , z 1 − α/ 2 ] we reject H 0 . � This is the same as checking µ 0 is in the confidence interval σ σ [ x + z α/ 2 √ n ; x + z 1 − α/ 2 √ n ] � Sometimes called a z -test (e.g. in MATLAB). 3/20 Dataanalyse - Kursusgang 3
χ 2 test Recap p -value Two sample tests Other goodness-of-fit checks Hypothesis test for µ = µ 0 with known σ � Suppose we have a hypothesis H 0 : µ = µ 0 ( µ 0 is a known number). Assuming H 0 is true we can calculate Z = X − µ 0 � ∼ N ( 0 , 1 ) σ 2 / n for a given data set. This is called the Z -statistic, and if Z / ∈ [ z α/ 2 , z 1 − α/ 2 ] we reject H 0 . � This is the same as checking µ 0 is in the confidence interval σ σ [ x + z α/ 2 √ n ; x + z 1 − α/ 2 √ n ] � Sometimes called a z -test (e.g. in MATLAB). 3/20 Dataanalyse - Kursusgang 3
χ 2 test Recap p -value Two sample tests Other goodness-of-fit checks Hypothesis test for µ = µ 0 with known σ � Suppose we have a hypothesis H 0 : µ = µ 0 ( µ 0 is a known number). Assuming H 0 is true we can calculate Z = X − µ 0 � ∼ N ( 0 , 1 ) σ 2 / n for a given data set. This is called the Z -statistic, and if Z / ∈ [ z α/ 2 , z 1 − α/ 2 ] we reject H 0 . � This is the same as checking µ 0 is in the confidence interval σ σ [ x + z α/ 2 √ n ; x + z 1 − α/ 2 √ n ] � Sometimes called a z -test (e.g. in MATLAB). 3/20 Dataanalyse - Kursusgang 3
χ 2 test Recap p -value Two sample tests Other goodness-of-fit checks When σ is unknown � Same same, but different! � Since σ 2 is unknown we use the estimator, S 2 ∼ σ 2 n − 1 χ 2 ( n − 1 ) , and we know T = X − µ � ∼ t ( n − 1 ) . (1) S 2 / n � The confidence interval is: s s [ x + t α/ 2 √ n ; x + t 1 − α/ 2 √ n ] (2) where t α/ 2 is the α/ 2 quantile in t ( n − 1 ) . � To test a hypothesis H 0 : µ = µ 0 we insert µ 0 in (1) and reject H 0 if the T -statistic is outside the interval [ t α/ 2 , t 1 − α/ 2 ] . This is the same as checking µ 0 is in the confidence interval (2). � Sometimes called a t -test. 4/20 Dataanalyse - Kursusgang 3
χ 2 test Recap p -value Two sample tests Other goodness-of-fit checks When σ is unknown � Same same, but different! � Since σ 2 is unknown we use the estimator, S 2 ∼ σ 2 n − 1 χ 2 ( n − 1 ) , and we know T = X − µ � ∼ t ( n − 1 ) . (1) S 2 / n � The confidence interval is: s s [ x + t α/ 2 √ n ; x + t 1 − α/ 2 √ n ] (2) where t α/ 2 is the α/ 2 quantile in t ( n − 1 ) . � To test a hypothesis H 0 : µ = µ 0 we insert µ 0 in (1) and reject H 0 if the T -statistic is outside the interval [ t α/ 2 , t 1 − α/ 2 ] . This is the same as checking µ 0 is in the confidence interval (2). � Sometimes called a t -test. 4/20 Dataanalyse - Kursusgang 3
χ 2 test Recap p -value Two sample tests Other goodness-of-fit checks When σ is unknown � Same same, but different! � Since σ 2 is unknown we use the estimator, S 2 ∼ σ 2 n − 1 χ 2 ( n − 1 ) , and we know T = X − µ � ∼ t ( n − 1 ) . (1) S 2 / n � The confidence interval is: s s [ x + t α/ 2 √ n ; x + t 1 − α/ 2 √ n ] (2) where t α/ 2 is the α/ 2 quantile in t ( n − 1 ) . � To test a hypothesis H 0 : µ = µ 0 we insert µ 0 in (1) and reject H 0 if the T -statistic is outside the interval [ t α/ 2 , t 1 − α/ 2 ] . This is the same as checking µ 0 is in the confidence interval (2). � Sometimes called a t -test. 4/20 Dataanalyse - Kursusgang 3
χ 2 test Recap p -value Two sample tests Other goodness-of-fit checks When σ is unknown � Same same, but different! � Since σ 2 is unknown we use the estimator, S 2 ∼ σ 2 n − 1 χ 2 ( n − 1 ) , and we know T = X − µ � ∼ t ( n − 1 ) . (1) S 2 / n � The confidence interval is: s s [ x + t α/ 2 √ n ; x + t 1 − α/ 2 √ n ] (2) where t α/ 2 is the α/ 2 quantile in t ( n − 1 ) . � To test a hypothesis H 0 : µ = µ 0 we insert µ 0 in (1) and reject H 0 if the T -statistic is outside the interval [ t α/ 2 , t 1 − α/ 2 ] . This is the same as checking µ 0 is in the confidence interval (2). � Sometimes called a t -test. 4/20 Dataanalyse - Kursusgang 3
χ 2 test Recap p -value Two sample tests Other goodness-of-fit checks When σ is unknown � Same same, but different! � Since σ 2 is unknown we use the estimator, S 2 ∼ σ 2 n − 1 χ 2 ( n − 1 ) , and we know T = X − µ � ∼ t ( n − 1 ) . (1) S 2 / n � The confidence interval is: s s [ x + t α/ 2 √ n ; x + t 1 − α/ 2 √ n ] (2) where t α/ 2 is the α/ 2 quantile in t ( n − 1 ) . � To test a hypothesis H 0 : µ = µ 0 we insert µ 0 in (1) and reject H 0 if the T -statistic is outside the interval [ t α/ 2 , t 1 − α/ 2 ] . This is the same as checking µ 0 is in the confidence interval (2). � Sometimes called a t -test. 4/20 Dataanalyse - Kursusgang 3
χ 2 test Recap p -value Two sample tests Other goodness-of-fit checks Example of hypothesis test � Suppose we measure the height of 100 people x 1 x 2 · · · x 100 178 cm 183 cm · · · 175 cm and calculate the estimate of the mean x = 178 and the estimate of the variance s 2 = 64 (i.e. s = 8). � We want to test H 0 : µ = 180 cm at level of significance α = 0 . 05. � Since t = 178 − 180 = − 2 . 5 , t 0 . 025 = − 1 . 98 , t 0 . 975 = 1 . 98 8 / 10 we reject H 0 . � What about with level of significance α = 0 . 01? In this case t 0 . 005 = − 2 . 63 , t 0 . 995 = 2 . 63 , and we cannot reject H 0 . 5/20 Dataanalyse - Kursusgang 3
χ 2 test Recap p -value Two sample tests Other goodness-of-fit checks Example of hypothesis test � Suppose we measure the height of 100 people x 1 x 2 · · · x 100 178 cm 183 cm · · · 175 cm and calculate the estimate of the mean x = 178 and the estimate of the variance s 2 = 64 (i.e. s = 8). � We want to test H 0 : µ = 180 cm at level of significance α = 0 . 05. � Since t = 178 − 180 = − 2 . 5 , t 0 . 025 = − 1 . 98 , t 0 . 975 = 1 . 98 8 / 10 we reject H 0 . � What about with level of significance α = 0 . 01? In this case t 0 . 005 = − 2 . 63 , t 0 . 995 = 2 . 63 , and we cannot reject H 0 . 5/20 Dataanalyse - Kursusgang 3
χ 2 test Recap p -value Two sample tests Other goodness-of-fit checks Example of hypothesis test � Suppose we measure the height of 100 people x 1 x 2 · · · x 100 178 cm 183 cm · · · 175 cm and calculate the estimate of the mean x = 178 and the estimate of the variance s 2 = 64 (i.e. s = 8). � We want to test H 0 : µ = 180 cm at level of significance α = 0 . 05. � Since t = 178 − 180 = − 2 . 5 , t 0 . 025 = − 1 . 98 , t 0 . 975 = 1 . 98 8 / 10 we reject H 0 . � What about with level of significance α = 0 . 01? In this case t 0 . 005 = − 2 . 63 , t 0 . 995 = 2 . 63 , and we cannot reject H 0 . 5/20 Dataanalyse - Kursusgang 3
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