# Central limit theorem question

State the Central Limit Theorem

The time spent by a factory worker A packing a box is a random variable with mean 3.5 minutes and standard deviation 1 minute. The times spent packing any two boxes are independent.

(i) What is the approximate probability that he packs 100 boxes in less than 6 hours?

(ii) What is the approximate probability that in 5.5 hours he will pack at least 90 and less than 100 boxes?

(iii) How long will he need to work for there to be 95% probability of packing 100 boxes?

(iv) A second worker B, does a similar job and the time it takes him to pack a box is a random variable with mean 3 minutes and standard deviation 1.2 minutes, independent of the time taken by A. The times spent by B packing any two boxes are independent. The two workers are assigned the job of packing 130 boxes each. Find the approximate probability that worker B will finish at least 50 minutes before worker A.

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#### Solution Preview

i) Central limit theorem

Let X_1,X_2,...,X_N be a set of N independent random variates and each X_i have an arbitrary probability distribution P(x_1,...,x_N) with mean mu_i and a finite variance sigma_i^2. Then the normal form variate

X_(norm)=(sum_(i==1)^(N)x_i-sum_(i==1)^(N)mu_i)/(sqrt(sum_(i==1)^(N)sigma_i^2)) (1)

has a limiting cumulative distribution function which approaches a normal distribution.

ii) Denote by X1,X2,...,X100 the life of the corresponding bulb, respectively. Then we know that

X1,X2,...,X100 are independent and follow the same distribution with mean mu=2000 hrs and standard deviation sigma=250hrs. Then the average life of the selected bulbs is

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#### Solution Summary

Solution provides detailed explanation, along with equations, on how to calculate central limit theorem.