There are three manufacturing business sectors for which the Tokutei gino visa is granted. 

They are:
素形材産業 formed and fabricated materials industries
産業機械製造業 industrial instruments manufacturing industries
電気・電子情報関連産業 Industries related to electrics, electronics and information.

For the above sectors, those who have mastered certain skills through technical training in Japan  suiccessuflly have chances to get the visa without taking tests.  The government has specified about 50 types of works.  Please see the chart and the sample of current certificates of the technical intern training program completion. 

For example, look at the first one on the chart: if you have received training for “Casting iron and article operation” and you have mastered the skill,
you could work for companies in “素形材産業formed and fabricated materials industries”
and  “産業機械製造業 industrial instruments manufactur-ing industries”,
but not in “電気・電子情報関連産業 Industries related to electrics, electronics and information.”

Another example is “紙器・段ボール箱製造 Carton box and corrugated card board box making.”  Even if you have received training for “段ボール箱製造作業Cardboard producing work,” you could not work for companies in manufacturing business sectors.

Have your certificate of technical raining at hand, and check if the skill you have mastered is on the list, and if so, which industries you could work for. 

(to be continued)

Based on the documents (Jan 21, 2019) by Ministry of Economy, Trade and Industry (METI) and oral explanation given by the officials from METI and from the Immigration Bureau.