A health club recently added tanning booths in addition to a weight room, pool,

Question

A health club recently added tanning booths in addition to a weight room, pool, and raquetball courts. For insurance purposes, an employee maintains careful records of the lenght of time each member spends in a tanning booth. The mean length of time spent tanning is 15 minutes with standard deviation 2 minutes. Consider a random sample of 35 members who use a tanning booth.

A. Find the distribution of the total time spent tanning by 35 members, T.

B. Find the probability that the total time spent tanning is between 8 and 9 hours.

C. If the total time spent tanning is more than 9.2 hours , an employee must work overtime. What is the probability that the employee works overtime on a day when 35 members tan.

D. Find a value t such that P(T >= t ) = 0.01

Explanation / Answer

A) here distribution of total time spent will be approximately normal with

mean =35*15=525

and std deviation =2*(35)1/2=11.832

b) as 8 hours =480 minute and 9 hours =540 minutes

P(480<X<540)=P((480-525)/11.832<Z<(540-525)/11.832)=P(-3.803<Z<1.268)=0.8976*0.0000=0.8975

c)here 9.2 hours =552 minutes

hence P(X>552)=1-P(X<552)=1-P(Z<(552-525)/11.382)=1-P(Z<2.282)=1-0.9888=0.0112

d) P(T>t)=0.01

P(T<t)=0.99

for above z=2.326

hence corresponding t =mean +z*std deviation =552.526

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