Considere a função !$ z=f(x,y)=6x^{1/2}y^{1/3} !$. Analisar o seguinte item:

Item 3 - A partir do ponto !$ (x_0,y_0)=(1,1) !$, se seguirmos a direção do vetor !$ (-1,1) !$, a função !$ f !$ irá decrescer para variações infinitesimais de !$ x !$ e !$ y !$.

Considere o seguinte sistema de equações lineares de oferta e demanda de trabalho das mulheres casadas. Neste sistema, as duas variáveis endógenas, salário (P) e quantidade de horas trabalhadas (Q), são determinadas pela educação (!$ X_1 !$), idade (!$ X_2 !$), número de filhos (!$ X_3 !$) e dois termos não-observáveis !$ (ε_{1i}, ε_{2i}) !$.

Demanda: !$ Q_i=\alpha_1P_i+\alpha_2X_{2i}+\alpha_3X_{3i}+ε_{1i} !$

Oferta: !$ P_i=\alpha_4Q_i+\alpha_5X_{1i}+\alpha_6X_{2i}+ε_{2i} !$

Temos uma amostra aleatória de N mulheres, !$ (Q_i,P_i,X_{1i},X_{2i},X_{3i}) !$, !$ i=1, \cdots, N !$.Vamos assumir que a matriz !$ [X_1,X_2,X_3] !$ tem posto completo.

Julgue o item abaixo:

Item 2 - Considerando somente a equação de oferta em sua forma reduzida, podemos obter !$ \hat{\alpha}_5 !$ em função somente de !$ \hat{∅}_1, \hat{∅}_2 !$ e !$ \hat{∅}_3 !$;

Based on your interpretation of the following texts, determine whether each statement is right or wrong.

Text 1

(from The Economist print edition, March 30^th – April 5^th 2013)

Excerpts from:

Climate science

A sensitive matter

The climate may be heating up less in response to greenhouse-gas emissions than was once thought. But that does not mean the problem is going away

OVER the past 15 years air temperatures at the Earth’s surface have been flat while greenhouse-gas emissions have continued to soar. The world added roughly 100 billion tonnes of carbon to the atmosphere between 2000 and 2010. That is about a quarter of all the CO₂ put there by humanity since 1750. And yet, as James Hansen, the head of NASA’s Goddard Institute for Space Studies, observes, “the five-year mean global temperature has been flat for a decade.”

Temperatures fluctuate over short periods, but this lack of new warming is a surprise. Ed Hawkins, of the University of Reading, in Britain, points out that surface temperatures since 2005 are already at the low end of the range of projections derived from 20 climate models (…). If they remain flat, they will fall outside the models’ range within a few years.

The mismatch between rising greenhouse-gas emissions and not-rising temperatures is among the biggest puzzles in climate science just now. It does not mean global warming is a delusion. Flat though they are, temperatures in the first decade of the 21st century remain almost 1°C above their level in the first decade of the 20th. But the puzzle does

need explaining. (…)

The insensitive planet

The term scientists use to describe the way the climate reacts to changes in carbon-dioxide levels is “climate sensitivity”. This is usually defined as how much hotter the Earth will get for each doubling of CO₂ concentrations. So-called equilibrium sensitivity, the commonest measure, refers to the temperature rise after allowing all feedback mechanisms to work (but without accounting for changes in vegetation and ice sheets).

Carbon dioxide itself absorbs infra-red at a consistent rate. For each doubling of CO₂ levels you get roughly 1°C of warming. A rise in concentrations from preindustrial levels of 280 parts per million (ppm) to 560ppm would thus warm the Earth by 1°C. If that were all there was to worry about, there would, as it were, be nothing to worry about. A 1°C rise could be shrugged off. But things are not that simple, for two reasons. One is that rising CO₂ levels directly influence phenomena such as the amount of water vapour (also a greenhouse gas) and clouds that amplify or diminish the temperature rise. This affects equilibrium sensitivity directly, meaning doubling carbon concentrations would produce more than a 1°C rise in temperature. The second is that other things, such as adding soot and other aerosols to the atmosphere, add to or subtract from the effect of CO₂. All serious climate scientists agree on these two lines of reasoning. But they disagree on the size of the change that is predicted.

(...)

The text implies that:

Item 1 - It was hotter in the first decade of the 20^th century than the first decade of the 21^st century;

Usando dados de uma amostra aleatória da população com 80.000 indivíduos, é estimada uma regressão pelo método de Mínimos Quadrados Ordinários. Os resultados dessa regressão são mostrados abaixo, em que os erros-padrão são mostrados entre parênteses:

[Para a resolução desta questão talvez lhe seja útil saber que se Z tem distribuição normal padrão, então P(|Z|>1,645)=0,10 e P(|Z|>1,96)=0,05]

Item 4 - É possível rejeitar, ao nível de significância de 5%, a hipótese nula de que o coeficiente associado a variável idade é igual a zero. A hipótese alternativa é que o coeficiente associado a variável idade é maior do que zero.