Hello. I am stuck on this problem. I am wondering if anyone could possibly point

Question

Hello. I am stuck on this problem. I am wondering if anyone could possibly point me in the right direction. Thanks. Here is the qestion.

At a certain fire station, false alarms are received at a mean rate of 0.2 per day. In a year, what is the probability that fewer than 52 false alarms are received? Note: You may need to use Excel to calculate the exact probabilities if the z score is not in the book's tables. (Consider 365 days a year. Round the standard deviation to 3 decimal places, z score to 2 decimals and final answer to 4 decimal places.)

Explanation / Answer

This is asking the probability that the mean per day of flase alarms is 52/365 = 0.142465753.

Note that for Poisson variables,

sigma = standard deviation = sqrt(u) = sqrt(0.2) = 0.447213595

We first get the z score for the critical value. As z = (x - u) sqrt(n) / s, then as          
          
x = critical value =    0.142465753      
u = mean =    0.2      
n = sample size =    52      
s = standard deviation =    0.447      
          
Thus,          
          
z = (x - u) * sqrt(n) / s =    -0.927711858 = -0.93      
          
Thus, using a table/technology, the left tailed area of this is          
          
P(z <   -0.93   ) = 0.1762 [ANSWER]

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