Question

**Central Limit Theorem**

The arrival of Blue Loop buses to the Penn State library stop
can be modeled as a Poisson process. Buses are scheduled to arrive
every 15 minutes. You and your friends are making observations at
the library stop during different times of the day. You and your
friends observe 53 Blue Loops arrivals, and you record the
inter-arrival times (i.e. the time between the arrival of the
n^{th} and the (n+1)^{st} bus in minutes.

**1. What is the probability that the average arrival time
that you and your friends observe for the Blue Loop is less than
15.7 minutes? (Use 3 decimal places)**

Answer #1

here for poisson distribution mean =15 and std deviation of one observation =sqrt(15.0)=3.873

for normal distribution z score =(X-μ)/σx | |

here mean= μ= | 15 |

std deviation =σ= | 3.8730 |

sample size =n= | 53 |

std error=σ_{x̅}=σ/√n= |
0.5320 |

**probability that the average arrival time that you and
your friends observe for the Blue Loop is less than 15.7
minutes** :

probability = | P(X<15.7) | = | P(Z<1.32)= | 0.907 |

( please try 0.906 if this comes wrong and revert)

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