Chapter I - Statistics II - Central Limit Theorem

This is a set of exercises for Statistics II, Chapter I — Central Limit Theorem, from the Lisbon Accounting and Business School (ISCAL–IPL).

Contents: We approach the Central Limit Theorem and use it to approximate other common distributions.

Questions

Question 1

The number of access to a certain website follows a Poisson distribution with an average rate of \(30\) per day.

  1. The probability (exact) that, in a a given day, the number of access is higher than 40 is exactly \(0.0323\). The approximate probability, by using the CLT, leads to an error of \(0.0049\), from the exact result.

  2. The probability that the number of accesses in a week (7 days) is between 200 and 220 (exclusive) is approximately:

Question 2

A complex system is formed by 100 parts that work indepedently. The probability that any of the components fails during the operating period is equal to \(0.1\).

  1. Knowing that the working of the system demands that at least 85 parts are operative, the approximate probability that the system works is equal to 0.9664.

  2. In the previous calculus, the use of the CLT demands only that the sample size, from a non Normal population is large enough.

Question 3

It is known that, in the TeleDigit, the probability that any phone call is to fill a complaint is equal to 0.05. Consider that the random experiment, consisting in picking a random sample of 20 calls, and count the ones that are complaints.

  1. The probability that in the sample there are between 2 and 5 (inclusive) is equal to 0.2639.

  2. The approximate probability that, in a random sample of 250 calls, at most 10 complaints are detected is:

Question 4

The number of machines ordered every day to a factory can be considered a random variable distributed Poisson with mean 9.5.

  1. The percentage of days in which more than 10 machines are ordered is equal to:

  2. Knowing that the exact probability of having being sent less than 30 machines in 4 work days is 0.07964, the approximation (by the CLT) error is equal to 0.00416.

Question 5

The lifetime, in years, of a lightning projector is a random variable. It is known that 20% of the projectors fail during the first year of use.

In a random sample of 500 projectors, the approximate probability that at most 80 fail during the first year is:

Question 6

Consider the time interval that passess between the arrival of two people to the reception desk of a hospital follows an exponential distribution with mean of 30 seconds.

  1. The approximate probability (by the CLT) that at most 40 people arrive to the reception desk of the hospital, during a time lapse of 18 minutes is 0.7734.

Question 7

In a gas station, each customer fills a random amount of gas \(X\), that it is known to follow a Uniform distribution in \([10,40]\) liters. Assume that, each day, the gas station has 300 customers.

The approximate probability that the daily (total) gas bought in that gas station is above 7410 liters of gas is equal to \(\Phi(0.6)\).

Question 8

A study about the sales in the sumermarket Superstore arrived to the conclusion that the daily rice demand (in Kg) is a random variable with mean 40Kg and standard deviation of 5Kg.

Having being ordered 14500 Kg of rice for sale in the next year, the probability that that stock satisfies the rice demand, in that year, is equal to 0.2643 (consider a year with 364 days).