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In a manufacturing plant, if the manager believes that more than [tex]$5\%$[/tex] of cans are dented, he will have the machines shut down for repair at great cost. Otherwise, he will let them continue to run.

To make the decision, the manager will inspect a random sample of 100 cans, then perform a test at the [tex]$\alpha=0.05$[/tex] significance level of [tex][tex]$H_0: p=0.05$[/tex][/tex] versus [tex]$H_a: p\ \textgreater \ 0.05$[/tex], where [tex]$p$[/tex] is the true proportion of all cans that are dented.

- Type I error:
- The manager finds convincing evidence that more than [tex][tex]$5\%$[/tex][/tex] of the cans are dented, when the true proportion really is [tex]$5\%$[/tex].

- Type II error:
- The manager does not find convincing evidence that more than [tex]$5\%$[/tex] of the cans are dented, when the true proportion really is more than [tex][tex]$5\%$[/tex][/tex].



Answer :

GinnyAnswer
Let's clarify what the errors in hypothesis testing mean in the context of this manufacturing problem.

Type I error:
- This occurs when we reject the null hypothesis [tex]\(H_0\)[/tex] when it is actually true. In this problem, the null hypothesis [tex]\(H_0\)[/tex] is that the true proportion of dented cans is [tex]\(5\%\)[/tex] (i.e., [tex]\(p = 0.05\)[/tex]). If the manager concludes that there is convincing evidence to suggest that more than [tex]\(5\%\)[/tex] of the cans are dented (thus rejecting [tex]\(H_0\)[/tex] in favor of [tex]\(H_a\)[/tex]), when in actual fact [tex]\(5\%\)[/tex] of the cans are dented, this is a Type I error.

So, for Type I error, the answer would be:
- The manager concludes there is convincing evidence that more than [tex]\(5 \%\)[/tex] of the cans are dented, when the true proportion really is [tex]\(5 \%\)[/tex].

Type II error:
- This occurs when we fail to reject the null hypothesis [tex]\(H_0\)[/tex] when it is actually false. In other words, the null hypothesis is that the proportion of dented cans is [tex]\(5\%\)[/tex] (i.e., [tex]\(p = 0.05\)[/tex]), but the true proportion is actually more than [tex]\(5\%\)[/tex]. If the manager fails to find convincing evidence that more than [tex]\(5\%\)[/tex] of the cans are dented (thus failing to reject [tex]\(H_0\)[/tex]), when the true proportion is actually more than [tex]\(5\%\)[/tex], this is a Type II error.

So, for Type II error, the answer would be:
- The manager fails to conclude there is convincing evidence that more than [tex]\(5\%\)[/tex] of the cans are dented, when the true proportion really is more than [tex]\(5 \%\)[/tex].

To summarize:
- Type I error: The manager concludes there is convincing evidence that more than [tex]\(5 \%\)[/tex] of the cans are dented, when the true proportion really is [tex]\(5 \%\)[/tex].
- Type II error: The manager fails to conclude there is convincing evidence that more than [tex]\(5 \%\)[/tex] of the cans are dented, when the true proportion really is more than [tex]\(5 \%\)[/tex].

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