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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

find the 15th term of the geometric sequince whose first is 18 and last term of 387
Jerwin Reply
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
virgelyn Reply
hmm well what is the answer
Abhi
how do they get the third part x = (32)5/4
kinnecy Reply
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
hmm
Abhi
is it a question of log
Abhi
🤔.
Abhi
I rally confuse this number And equations too I need exactly help
salma
But this is not salma it's Faiza live in lousvile Ky I garbage this so I am going collage with JCTC that the of the collage thank you my friends
salma
Commplementary angles
Idrissa Reply
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
hii
Uday
hi
salma
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
kkk nice
Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
how do you translate this in Algebraic Expressions
linda Reply
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
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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