In a final exam, there are 300 students and one-third of them bring a water bott

ID: 3131223 • Letter: I

Question

In a final exam, there are 300 students and one-third of them bring a water bottle. Suppose that for those who bring a water bottle, the chance that each of them leaves behind his/her water bottle in the examination room after the exam is 0.06.

Part a) What are the mean and standard deviation of the total number of water bottles left behind after the exam? Give your answers to 2 decimal places.

Part b) Using an appropriate approximation method, find the probability that 5 or less water bottles are left behind. Carry your answers from the previous part to at least 6 decimal places in your calculation here. Give your final answer to 4 decimal places.

Part c) Using an appropriate approximation method, at most how many water bottles will be left behind 84.13% of the time? Give your answer to the nearest integer.

Explanation / Answer

Part a) What are the mean and standard deviation of the total number of water bottles left behind after the exam? Give your answers to 2 decimal places.

n=300/3=100 p=0.06

Expectation = np = 6

Variance = np(1 - p) = 5.64

Standard deviation = 2.37

With continuity correction, z value for 5 or less, z=(5.5-6)/ 2.374868 =-0.21

P( x <=5) = P( z < -0.21) =0.4168

Part c) Using an appropriate approximation method, at most how many water bottles will be left behind 84.13% of the time? Give your answer to the nearest integer.

Z value for 84.13% =1

X=6+1*2.37 =8.37

Answer: 8

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