a$.**$Causal or Correlational Study$**$

$-$ This would be a $**$causal study$**.$ The machine tool worker aims to prove that the exhaust fumes cause low efficiency among workers, indicating a direct cause$-$and$-$effect relationship rather than just an association.

b$.**$Exploratory$,$ Descriptive, or Prescriptive Study$**$

$-$ This would be a $**$prescriptive study$**.$ The goal is to provide evidence that can be used to justify changes to improve worker efficiency, such as reducing or eliminating exhaust fumes in the workshop. It moves beyond simply describing or exploring the phenomenon to recommending specific actions.

c$.**$Study Design$**$

$-$ The study would likely be designed as a $**$field experiment$**.$ This design allows the researcher to manipulate the level of exhaust fumes in the actual workshop environment $($the field$)$ and observe the effects on worker efficiency. This approach maintains the real$-$world context while allowing for control over the variable of interest $($exhaust fumes$).$

d$.**$Unit of Analysis$**$

$-$ The unit of analysis would be the $**$individual workers$**.$ The research aims to measure the impact of exhaust fumes on each worker's efficiency, making the individual worker the focus of analysis.

e$.**$Cross$-$sectional or Longitudinal Study$**$

$-$ This would be a $**$cross$-$sectional study$**$ if the researcher measures the workers' efficiency at a single point in time after manipulating the exhaust fumes. However, if the researcher measures the effects of exhaust fumes on workers' efficiency over an extended period to observe changes and trends, it would be a $**$longitudinal study$**.$ Given the need to prove causality and observe the impact over time, a longitudinal approach would likely be more appropriate, as it provides stronger evidence of causality by showing changes over time.