Air traffic controllers have the crucial task of ensuring that aircraft don't collide. To do this, they must quickly discern when two planes are about to enter the same air space at the same time. They are aided by video display panels that track the aircraft in their sector and alert the controller when two flight paths are about to converge. The display panel currently in use has a mean "alert time" of $15$ seconds. $($The alert time is the time elapsing between the instant when two aircraft enter into a collision course and when a controller initiates a call to reroute the planes.$)$ According to Ralph Rudd, a supervisor of air traffic controllers at the Greater Cincinnati International Airport, a new display panel has been developed that uses artificial intelligence to project a plane's current flight path into the future. This new panel provides air traffic controllers with an earlier warning that a collision is likely. It is hoped that the mean "alert time," $\backslash$mu $,$ for the new panel is less than $8$ seconds. In order to test the new panel, $15$ randomly selected air traffic controllers are trained to use the panel and their alert times for a simulated collision course are recorded. The sample alert times $($in seconds$)$ are: $7\mathrm{.}0,6\mathrm{.}3,8\mathrm{.}0,8\mathrm{.}7,9\mathrm{.}5,9\mathrm{.}0,9\mathrm{.}5,9\mathrm{.}0,7\mathrm{.}5,7\mathrm{.}8,8\mathrm{.}3,5\mathrm{.}8,6\mathrm{.}6,7\mathrm{.}1,7\mathrm{.}3.$

$(1)$ Using the fact that x$$

x

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$=7\mathrm{.}9$ and s $=1\mathrm{.}223,$ find a $95$ percent confidence interval for the population mean alert time, $\backslash$mu $,$ for the new panel. $($Round your answers to $3$ decimal places.$)$