Step 1: Understanding the reaction.
When t-butyl bromide (C\(_4\)H\(_9\)Br) is treated with alcoholic ammonia (NH\(_3\)), the ammonia replaces the bromine atom (via nucleophilic substitution) to form an amine. However, due to the structure of t-butyl bromide, a double bond is formed through elimination. The resulting compound will be an alkene, specifically isobutene (CH\(_3\)-C=CH\(_2\)).
Step 2: Analyzing the options.
(A) CH\(_3\)-C-CH\(_3\)-NHBr: Incorrect. This is a product of nucleophilic substitution but not elimination.
(B) CH\(_3\)-C-CH\(_2\)-NH\(_2\)Br: Incorrect. This compound contains an amine group, but this is not the result of the reaction described.
(C) CH\(_3\)-C=CH\(_2\): Correct. This is the product of elimination in the presence of alcoholic ammonia, resulting in isobutene.
(D) CH\(_3\)-C-CH\(_3\)-NH\(_2\): Incorrect. This product would be obtained if an amine formed by nucleophilic substitution without elimination, which is not the case here.
Step 3: Conclusion.
The correct product formed is CH\(_3\)-C=CH\(_2\), corresponding to option (C).