# Various probability problems

Please find attached problem.

(a) A five-gear assembly is put together with spacers between the gears. The mean thickness of the gears is 5.030 cm with a standard deviation of 0.008cm. The mean thickness of the spacers is 0.140 cm with the standard deviation of 0.005 cm. Find the mean and standard deviation of the assembled units consisting of five randomly selected gears and four randomly selected spacers.

(b) Four athletes specialise in running 200m, 200m, 400m and 800m respectively. They train as a team for the 1600m-medley relay race. During training their mean times for their respective distances are 23.9 seconds, 24.1 seconds, 53.6 seconds and 2 minutes 7.4 seconds, and the corresponding standard deviations are 0.3 seconds, 0.3 seconds, 0.8 seconds and 1.8 seconds. Assuming that their individual times are normally and independently distributed, estimate the mean and standard deviation of the times in which the team covers 1600m. Also estimate that the probability that, on any particular occasion, the time will be 3 minutes 45 seconds or less.

(c ) A sample size of 100 was taken from the population with:

What is the probability that the sample mean will be:

1) Less than 30.0
2) Less than or equal to 28.0
3) More than or equal to 29.5
4) Between 28.0 and 29.0?

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