$($a$)$ Construct a $95\%$ confidence interval for the mean lifetime of this type of drill. Use tables or technology. Round your answers to at least two decimal places. The confidence interval is $(,)($b$)$ The manufacturer of the drills claims that the mean lifetime is greater than $\mathrm{.}13$ Does this confidence interval contradict this claim? $,$ Choose One $-$ Yes$/$No the confidence interval contains some$/$ does not contain any values greater than $13,$ so the claim is$/$ is not contradicted. $($c$)$ How large would the sample need to be so that a $95\%$ confidence interval would have a margin of error of $1\mathrm{.}0?$

The sample size would need to be: