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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

An investment account was opened with an initial deposit of $9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years?
Kala Reply
lim x to infinity e^1-e^-1/log(1+x)
given eccentricity and a point find the equiation
Moses Reply
12, 17, 22.... 25th term
Alexandra Reply
12, 17, 22.... 25th term
Akash
College algebra is really hard?
Shirleen Reply
Absolutely, for me. My problems with math started in First grade...involving a nun Sister Anastasia, bad vision, talking & getting expelled from Catholic school. When it comes to math I just can't focus and all I can hear is our family silverware banging and clanging on the pink Formica table.
Carole
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
Jeannette has $5 and $10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.
August Reply
What is the expressiin for seven less than four times the number of nickels
Leonardo Reply
How do i figure this problem out.
how do you translate this in Algebraic Expressions
linda Reply
why surface tension is zero at critical temperature
Shanjida
I think if critical temperature denote high temperature then a liquid stats boils that time the water stats to evaporate so some moles of h2o to up and due to high temp the bonding break they have low density so it can be a reason
s.
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
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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