Oti93
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Times can be modelled by X~N(65,20)

Given that the times are independent from each day, determine that in a period of 6 days;

the time to complete each crossword is less than 60 minutes

her mean completion time is less than 60 minutes

completely forgotten how to do these, so any help appreciated.
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Oti93
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anybody?
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Morbo
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(Original post by Oti93)
Times can be modelled by X~N(65,20)

Given that the times are independent from each day, determine that in a period of 6 days;

(1) the time to complete each crossword is less than 60 minutes

(2) her mean completion time is less than 60 minutes

completely forgotten how to do these, so any help appreciated.
I assume that "she" does one crossword each day, and these questions want you to determine the probability of these outcomes.

The distribution tells you that she completes these crosswords with a mean time of 65 minutes and a variance of 20 minutes.

Over 6 days, she does 6 crosswords i.e. you are taking a random sample of 6 from this distribution.

Problem (1)
I think this is the probability of getting 6 times that are all less than 60 minutes. The probability of getting 1 time that is less than 60 minutes is given by the distribution as:

P_{t<60} = P(t < 60), where P(t < 60) is given by the cumulative normal distribution function, or can be looked up in normal distribution tables after converting to the standard distribution, commonly written as Z.

Since we are now looking at a problem with a fixed probability, p = P_{t<60} and a discrete number of trials, n = 6, it becomes one of a binomial distribution, where the general binomial distribution is:
P = B(n,p).

However, since the problem only asks for the probabiliity where all trials are successful (i.e. all crosswords are completed in under 60mins), the calculation is simple, it is just:

P_{(1)} = p^n


Problem (2)
Use the Central Limit Theorem. The probability distribution of a random sample has the same mean as the population, but its variance is modified. Find out how, and this gives you a new distribution from which you can calculate P'(t<60).
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