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η Y x = x Y Y x = β 1 x Y . MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4TdG2aaSbaaSqaaiaadMfacaWG4baabeaakiabg2da9maalaaabaGaamiEaaqaaiaadMfaaaWaaSaaaeaacqGHciITcaWGzbaabaGaeyOaIyRaamiEaaaacqGH9aqpcqaHYoGydaWgaaWcbaGaaGymaaqabaGcdaWcaaqaaiaadIhaaeaacaWGzbaaaiaac6caaaa@478D@

Clearly, researchers need to choose the levels of Y and x at which to report this elasticity; it is traditional to calculate the elasticity at the means. Thus, economists typically report

η Y x = β 1 x ¯ Y ¯ . MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4TdG2aaSbaaSqaaiaadMfacaWG4baabeaakiabg2da9iabek7aInaaBaaaleaacaaIXaaabeaakmaalaaabaGabmiEayaaraaabaGabmywayaaraaaaiaac6caaaa@4015@

Constant elasticities

Consider the following demand equation:

q = α p β e ε , MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyCaiabg2da9iabeg7aHjaadchadaahaaWcbeqaaiabgkHiTiabek7aIbaakiaadwgadaahaaWcbeqaaiabew7aLbaakiaacYcaaaa@40C0@

where q is the quantity demanded, p is the price the good is sold at, α , β > 0 , MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeg7aHjaacYcacqaHYoGycqGH+aGpcaaIWaGaaiilaaaa@3C4B@ and ε MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdugaaa@379B@ is an error term. The price elasticity of demand is given by

η q p = p q q p = p α p β e ε ( β α p β 1 e ε ) = β . MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4TdG2aaSbaaSqaaiaadghacaWGWbaabeaakiabg2da9maalaaabaGaamiCaaqaaiaadghaaaWaaSaaaeaacqGHciITcaWGXbaabaGaeyOaIyRaamiCaaaacqGH9aqpdaWcaaqaaiaadchaaeaacqaHXoqycaWGWbWaaWbaaSqabeaacqGHsislcqaHYoGyaaGccaWGLbWaaWbaaSqabeaacqaH1oqzaaaaaOWaaeWaaeaacqGHsislcqaHYoGycqaHXoqycaWGWbWaaWbaaSqabeaacqGHsislcqaHYoGycqGHsislcaaIXaaaaOGaamyzamaaCaaaleqabaGaeqyTdugaaaGccaGLOaGaayzkaaGaeyypa0JaeyOeI0IaeqOSdiMaaiOlaaaa@5DDA@

In other words, this demand curve has a constant price elasticity of demand equal to β . MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaeqOSdiMaaiOlaaaa@3933@ Moreover, we can convert the estimation of this equation into a linear regression by taking the natural logarithm of both sides of (5) to get ln q = ln α β ln p + ε . MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac6gacaWGXbGaeyypa0JaciiBaiaac6gacqaHXoqycqGHsislcqaHYoGyciGGSbGaaiOBaiaadchacqGHRaWkcqaH1oqzcaGGUaaaaa@45F8@

The logit equation and the quasi-elasticity

It is not appropriate to use the normal formula for an elasticity with (3) because the dependent variable is itself a number without units between 0 and 1. As an alternative it makes more sense to calculate the quasi-elasticity , which is defined as:

η ( x ) = x Pr ( x ) x . MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4TdG2aaeWaaeaacaWG4baacaGLOaGaayzkaaGaeyypa0JaamiEamaalaaabaGaeyOaIyRaciiuaiaackhadaqadaqaaiaadIhaaiaawIcacaGLPaaaaeaacqGHciITcaWG4baaaiaac6caaaa@4505@

Since

ln ( Pr ( x i ) 1 Pr ( x i ) ) = β 0 + β 1 x i + ε i , MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac6gadaqadaqaamaalaaabaGaciiuaiaackhadaqadaqaaiaadIhadaWgaaWcbaGaamyAaaqabaaakiaawIcacaGLPaaaaeaacaaIXaGaeyOeI0IaciiuaiaackhadaqadaqaaiaadIhadaWgaaWcbaGaamyAaaqabaaakiaawIcacaGLPaaaaaaacaGLOaGaayzkaaGaeyypa0JaeqOSdi2aaSbaaSqaaiaaicdaaeqaaOGaey4kaSIaeqOSdi2aaSbaaSqaaiaaigdaaeqaaOGaamiEamaaBaaaleaacaWGPbaabeaakiabgUcaRiabew7aLnaaBaaaleaacaWGPbaabeaakiaacYcaaaa@538D@

we can calculate this elasticity as follows:

( ln ( Pr ( x i ) 1 Pr ( x i ) ) ) x = β 1 . MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacqGHciITdaqadaqaaiGacYgacaGGUbWaaeWaaeaadaWcaaqaaiGaccfacaGGYbWaaeWaaeaacaWG4bWaaSbaaSqaaiaadMgaaeqaaaGccaGLOaGaayzkaaaabaGaaGymaiabgkHiTiGaccfacaGGYbWaaeWaaeaacaWG4bWaaSbaaSqaaiaadMgaaeqaaaGccaGLOaGaayzkaaaaaaGaayjkaiaawMcaaaGaayjkaiaawMcaaaqaaiabgkGi2kaadIhaaaGaeyypa0JaeqOSdi2aaSbaaSqaaiaaigdaaeqaaOGaaiOlaaaa@4FB0@

Focusing on the left-hand-side, we get:

1 Pr ( x i ) Pr ( x i ) ( 1 Pr ( x i ) ) Pr ( x i ) x + Pr ( x i ) Pr ( x i ) x ( 1 Pr ( x i ) ) 2 = β 1 MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@716C@

or

1 Pr ( x i ) ( 1 Pr ( x i ) ) Pr ( x i ) x = β 1 MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaaIXaaabaGaciiuaiaackhadaqadaqaaiaadIhadaWgaaWcbaGaamyAaaqabaaakiaawIcacaGLPaaadaqadaqaaiaaigdacqGHsislciGGqbGaaiOCamaabmaabaGaamiEamaaBaaaleaacaWGPbaabeaaaOGaayjkaiaawMcaaaGaayjkaiaawMcaaaaadaWcaaqaaiabgkGi2kGaccfacaGGYbWaaeWaaeaacaWG4bWaaSbaaSqaaiaadMgaaeqaaaGccaGLOaGaayzkaaaabaGaeyOaIyRaamiEaaaacqGH9aqpcqaHYoGydaWgaaWcbaGaaGymaaqabaaaaa@51B8@

or

Pr ( x i ) x = β 1 Pr ( x i ) ( 1 Pr ( x i ) ) . MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacqGHciITciGGqbGaaiOCamaabmaabaGaamiEamaaBaaaleaacaWGPbaabeaaaOGaayjkaiaawMcaaaqaaiabgkGi2kaadIhaaaGaeyypa0JaeqOSdi2aaSbaaSqaaiaaigdaaeqaaOGaciiuaiaackhadaqadaqaaiaadIhadaWgaaWcbaGaamyAaaqabaaakiaawIcacaGLPaaadaqadaqaaiaaigdacqGHsislciGGqbGaaiOCamaabmaabaGaamiEamaaBaaaleaacaWGPbaabeaaaOGaayjkaiaawMcaaaGaayjkaiaawMcaaiaac6caaaa@51A9@

Thus, we see from (6) that the quasi-elasticity is given by:

η ( x i ) = β 1 x i Pr ( x i ) ( 1 Pr ( x i ) ) . MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4TdG2aaeWaaeaacaWG4bWaaSbaaSqaaiaadMgaaeqaaaGccaGLOaGaayzkaaGaeyypa0JaeqOSdi2aaSbaaSqaaiaaigdaaeqaaOGaamiEamaaBaaaleaacaWGPbaabeaakiGaccfacaGGYbWaaeWaaeaacaWG4bWaaSbaaSqaaiaadMgaaeqaaaGccaGLOaGaayzkaaWaaeWaaeaacaaIXaGaeyOeI0IaciiuaiaackhadaqadaqaaiaadIhadaWgaaWcbaGaamyAaaqabaaakiaawIcacaGLPaaaaiaawIcacaGLPaaacaGGUaaaaa@4FD1@

The quasi-elasticity measures the percentage point change in the probability due to a 1 percent increase of x . Notice that it is dependent on what value of x it is evaluated at. It is usual to evaluate (8) at the mean of x . Thus, the quasi-elasticity at the mean of x is:

η ( x ¯ ) = β 1 x ¯ Pr ( x ¯ ) ( 1 Pr ( x ¯ ) ) , MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4TdG2aaeWaaeaaceWG4bGbaebaaiaawIcacaGLPaaacqGH9aqpcqaHYoGydaWgaaWcbaGaaGymaaqabaGcceWG4bGbaebaciGGqbGaaiOCamaabmaabaGabmiEayaaraaacaGLOaGaayzkaaWaaeWaaeaacaaIXaGaeyOeI0IaciiuaiaackhadaqadaqaaiqadIhagaqeaaGaayjkaiaawMcaaaGaayjkaiaawMcaaiaacYcaaaa@4B9F@

where

Pr ( x ¯ ) = e β 0 + β 1 x ¯ 1 + e β 0 + β 1 x ¯ . MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiuaiaackhadaqadaqaaiqadIhagaqeaaGaayjkaiaawMcaaiabg2da9maalaaabaGaamyzamaaCaaaleqabaGaeqOSdi2aaSbaaWqaaiaaicdaaeqaaSGaey4kaSIaeqOSdi2aaSbaaWqaaiaaigdaaeqaaSGabmiEayaaraaaaaGcbaGaaGymaiabgUcaRiaadwgadaahaaWcbeqaaiabek7aInaaBaaameaacaaIWaaabeaaliabgUcaRiabek7aInaaBaaameaacaaIXaaabeaaliqadIhagaqeaaaaaaGccaGGUaaaaa@4E40@

Hypothesis testing

The researcher using the logit model (and any regression estimated by ML) has three choices when constructing tests of hypotheses about the unknown parameter estimates—(1) the Wald test statistic, (2) the likelihood ratio test, or (3) the Lagrange Multiplier test. We consider them in turn.

The wald test

The Wald test is the most commonly used test in econometric models. Indeed, it is the one that most statistics students learn in their introductory courses. Consider the following hypothesis test:

H 0 : β 1 = β H A : β 1 β . MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaqGibWaaSbaaSqaaiaabcdaaeqaaOGaaeOoaiaabccacqaHYoGydaWgaaWcbaGaaGymaaqabaGccqGH9aqpcqaHYoGyaeaacaqGibWaaSbaaSqaaiaabgeaaeqaaOGaaeOoaiaabccacqaHYoGydaWgaaWcbaGaaGymaaqabaGccqGHGjsUcqaHYoGycaGGUaaaaaa@4818@

Quite often in these test researchers are interested in the case when β = 0 MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdiMaeyypa0JaaGimaaaa@3955@ —i.e., in testing if the independent variable’s estimated parameter is statistically different from zero. However, β MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdigaaa@3794@ can be any value. Moreover, this test can be used to test multiple restrictions on the slope parameters for multiple independent variables. In the case of a hypothesis test on a single parameter, the t-ratio is the appropriate test statistic. The t-statistic is given by

t = β i β s .e . ( β i ) ~ t n k 1 , MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaiabg2da9maalaaabaGafqOSdiMbambadaWgaaWcbaGaamyAaaqabaGccqGHsislcqaHYoGyaeaacaqGZbGaaeOlaiaabwgacaqGUaWaaeWaaeaacuaHYoGygaWeamaaBaaaleaacaWGPbaabeaaaOGaayjkaiaawMcaaaaacaGG+bGaamiDamaaBaaaleaacaWGUbGaeyOeI0Iaam4AaiabgkHiTiaaigdaaeqaaOGaaiilaaaa@4C70@

where k is the number of parameters in the mode that are estimated. The F-statistic is the appropriate test statistic when the null hypothesis has restrictions on multiple parameters. See Cameron and Trivedi (2005: 224-231) for more detail on this test. According to Hauck and Donner (1977) the Wald test may exhibit perverse behavior when the sample size is small. For this reason this test must be used with some care.

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Econometrics for honors students. OpenStax CNX. Jul 20, 2010 Download for free at http://cnx.org/content/col11208/1.2
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