Question

Can you please show the steps and mention the name of the theorems you are using thanks!

Suppose that an electronic temperature sensor for fluids (including plasmas, gases, and liquids) obtains temperatures by randomly testing the fluid at various locations to produce an average reported temperature. Suppose further that a fluid of known (theoretical) mean temperature 180 degrees Kelvin with standard deviation of 6 degrees Kelvin is being studied to calibrate the sensor. a) [5] If the sensor randomly samples the fluid at 25 locations, what is the probability that the sensor reports a temperature between 179 and 181 degrees Kelvin? b) [5] At how many locations should the sensor sample the fluid in order to ensure that the probability in part (a) is at least 0.99?

Answer 1

Solution: Given that μ = 180, σ = 6

a) for n = 25
P(179 < X < 181) = P((179-180)/(6/sqrt(25)) < (x-μ)/(σ/sqrt(n)) < (181-180)/(6/sqrt(25)))
= P(-0.8333 < Z < 0.8333)
= P(Z < 0.8333) − P(Z < −0.8333)
= 0.7967 - 0.2033
= 0.5934 (rounded)

=> P(Z < 0.8333) can be found by using the following standard normal table:

b) P(Z >= z) = 0.99 => z = 2.33

X = μ + z*σ
= 180 + 2.33*6
= 193.98