Pesquisadores desenvolveram um modelo de regressão linear simples para explicar a produtividade de f...

Question

Pesquisadores desenvolveram um modelo de regressão linear simples para explicar a produtividade de funcionários em função de horas de capacitação. O parâmetro β₁ foi estimado em β̂₁ = 1,5 com erro padrão de 0,5, utilizando 16 observações. Os pesquisadores aplicaram um teste de hipótese unilateral à direita, com α=0,05.

Com base nesse cenário e sobre o teste de hipótese realizado e a significância do coeficiente angular, analise as afirmações a seguir:

I. O valor da estatística t é maior que o valor crítico da distribuição t para 14 graus de liberdade e α=0,05, justificando a rejeição de H₀ .

II. O valor da estatística t é 3,0, indicando que o coeficiente estimado está a três erros padrões de distância do valor hipotético.

III. O valor crítico da t-student neste caso é aproximadamente 2,145, que não é ultrapassado pela estatística t, indicando que H₀ não deve ser rejeitada.

IV. O teste foi realizado com 14 graus de liberdade, pois se subtrai dois parâmetros do total da amostra no modelo de regressão linear simples.

Está correto o que se afirma em:

Calculadora Univesp Gabarito · Accepted Answer

A resposta correta é a alternativa B: I, II e IV, apenas.

Marcone Magalhaes · Answer

**Alternativa (B)** - I, II e IV, apenas.

### Introdução
A questão aborda o teste de hipóteses para o coeficiente angular (β₁) em um modelo de regressão linear simples. Este é um tópico fundamental em estatística aplicada, especialmente para inferir se uma variável preditora (horas de capacitação) tem um impacto estatisticamente significativo na variável resposta (produtividade de funcionários). A compreensão de conceitos como estatística t, graus de liberdade, valor crítico e regra de decisão é crucial para interpretar corretamente os resultados de um modelo de regressão.

### Desenvolvimento
O modelo de regressão linear simples é dado por Y = β₀ + β₁X + ε, onde Y é a variável dependente, X é a variável independente, β₀ é o intercepto, β₁ é o coeficiente angular e ε é o termo de erro aleatório.
Para testar a significância do coeficiente angular (β₁), geralmente se formula um teste de hipóteses:
*   **Hipótese Nula (H₀):** β₁ = 0 (Não há relação linear entre X e Y, ou X não tem efeito sobre Y).
*   **Hipótese Alternativa (H₁):** β₁ > 0 (Há uma relação linear positiva, ou X tem um efeito positivo sobre Y).
Neste caso, a questão especifica um teste unilateral à direita.

A estatística de teste utilizada é a distribuição t de Student, calculada como:
t = (β̂₁ - β₁₀) / SE(β̂₁)
Onde:
*   β̂₁ é a estimativa do coeficiente angular.
*   β₁₀ é o valor hipotético de β₁ sob a hipótese nula (neste caso, 0).
*   SE(β̂₁) é o erro padrão da estimativa de β̂₁.

Os **graus de liberdade (df)** para um modelo de regressão linear simples são calculados como n - k - 1, onde n é o número de observações e k é o número de variáveis preditoras (neste caso, k=1 para a variável 'horas de capacitação'). Portanto, df = n - 2.

A decisão de rejeitar ou não a hipótese nula é baseada na comparação da estatística t calculada com um **valor crítico (t_crítico)** da distribuição t, obtido a partir de uma tabela t-Student, com base nos graus de liberdade e no nível de significância (α). Para um teste unilateral à direita, se t > t_crítico, rejeita-se H₀.

### Análise

Vamos analisar cada afirmação com base nos dados fornecidos: β̂₁ = 1,5, SE(β̂₁) = 0,5, n = 16 observações, α = 0,05 (teste unilateral à direita).

1.  **Cálculo dos Graus de Liberdade (df):**
    df = n - 2 = 16 - 2 = 14.

2.  **Cálculo da Estatística t:**
    t = (β̂₁ - 0) / SE(β̂₁) = 1,5 / 0,5 = 3,0.

3.  **Determinação do Valor Crítico (t_crítico):**
    Para um teste unilateral à direita com df = 14 e α = 0,05, consultando uma tabela da distribuição t-Student, o valor crítico é aproximadamente 1,761.

Agora, avaliemos as afirmações:

| Afirmação | Análise Detalhada | Status |
| :-------- | 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| β₀ (intercept)    | α (nível de significância) | n (número de observações) | β̂₁ (coeficiente estimado) | SE(β̂₁) (erro padrão) | Graus de Liberdade (df) | H₀           | H₁           | t-calculado        | t-crítico para α=0,05 (teste unilateral à direita) com df=14 | Decisão              |
| :---------------- | :------------------------- | :------------------------ | :------------------------ | :-------------------- | :---------------------- | :----------- | :----------- | :----------------- | :------------------------------------------------------ | :------------------- |
| Não informado     | 0,05                       | 16                        | 1,5                       | 0,5                   | 14                      | β₁ = 0       | β₁ > 0       | 3,0                | 1,761                                                   | Rejeitar H₀          |

*   **Afirmação I:** O valor da estatística t é maior que o valor crítico da distribuição t para 14 graus de liberdade e α=0,05, justificando a rejeição de H₀.
    *   t_calculado = 3,0.
    *   t_crítico (df=14, α=0,05, unilateral à direita) ≈ 1,761.
    *   Como 3,0 > 1,761, a estatística t é de fato maior que o valor crítico, levando à rejeição de H₀.
    *   **A afirmação I está correta.**

*   **Afirmação II:** O valor da estatística t é 3,0, indicando que o coeficiente estimado está a três erros padrões de distância do valor hipotético.
    *   t_calculado = 1,5 / 0,5 = 3,0.
    *   O valor da estatística t representa quantas unidades de erro padrão o coeficiente estimado (β̂₁) está distante do valor hipotético sob H₀ (neste caso, 0). Portanto, 3,0 significa "3 erros padrões de distância".
    *   **A afirmação II está correta.**

*   **Afirmação III:** O valor crítico da t-student neste caso é aproximadamente 2,145, que não é ultrapassado pela estatística t, indicando que H₀ não deve ser rejeitada.
    *   O valor crítico para df=14 e α=0,05 (unilateral à direita) é aproximadamente 1,761. O valor 2,145 seria para um teste bilateral com α=0,05 (ou unilateral com α=0,025) e 14 graus de liberdade.
    *   Mesmo que o valor crítico fosse 2,145, a estatística t calculada (3,0) *ultrapassaria* 2,145 (3,0 > 2,145), levando à rejeição de H₀.
    *   A conclusão de que H₀ não deve ser rejeitada é incorreta, seja com o valor crítico correto ou com o valor crítico sugerido.
    *   **A afirmação III está incorreta.**

*   **Afirmação IV:** O teste foi realizado com 14 graus de liberdade, pois se subtrai dois parâmetros do total da amostra no modelo de regressão linear simples.
    *   Os graus de liberdade para um modelo de regressão linear simples são df = n - 2.
    *   Para n=16 observações, df = 16 - 2 = 14.
    *   Os dois parâmetros subtraídos correspondem à estimativa de β₀ (intercepto) e β₁ (coeficiente angular).
    *   **A afirmação IV está correta.**

As afirmações corretas são I, II e IV.

### Conclusão
O teste de hipóteses para o coeficiente angular (β₁) em regressão linear simples avalia a significância da relação entre a variável preditora e a variável resposta. Neste cenário, a estatística t calculada foi de 3,0, com 14 graus de liberdade. Dado um nível de significância de 0,05 para um teste unilateral à direita, o valor crítico da distribuição t é aproximadamente 1,761. Como 3,0 é maior que 1,761, a hipótese nula de que β₁ é zero é rejeitada, indicando que as horas de capacitação têm um efeito positivo e estatisticamente significativo na produtividade. A afirmação III está incorreta por apresentar um valor crítico inadequado para as condições do teste e uma conclusão errônea.

Alternativa **(B)**.

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