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Statistics S1

Hi,

I did this question from a past paper on the normal distribution and wasn't sure how to do it, and after looking at the mark scheme I still can't seem to understand how to do it, so was hoping someone can help.

A health club lets members use, on each visit, its facilities for as long as they wish. The club's records suggest that the length of a visit can be modelled by a normal distribtion with mean 90 minutes and standard deviation 41.587. Then club introduce a closing time of 10.00pm.

Tara arrives at the club at 8.00pm Explain whether or not this distribution is still a suitable model for the length of her visit.

The mark scheme states 90 + 3σ= 215 => 6.25 pm for latest arrival 90 + 2σ = 173.3 => 7.07 pm for latest arrival
Therefore this normal distribution is not suitable. (Based on 2σ/3σ rule)

Can someone please explain to me how what the mark scheme shows means that the distribution is not suitable, as I can't figure it out. Also, what is the '2σ/3σ rule'? Thank you very much for your help!
Original post by Jacob4
Hi,

I did this question from a past paper on the normal distribution and wasn't sure how to do it, and after looking at the mark scheme I still can't seem to understand how to do it, so was hoping someone can help.

A health club lets members use, on each visit, its facilities for as long as they wish. The club's records suggest that the length of a visit can be modelled by a normal distribtion with mean 90 minutes and standard deviation 41.587. Then club introduce a closing time of 10.00pm.

Tara arrives at the club at 8.00pm Explain whether or not this distribution is still a suitable model for the length of her visit.

The mark scheme states 90 + 3σ= 215 => 6.25 pm for latest arrival 90 + 2σ = 173.3 => 7.07 pm for latest arrival
Therefore this normal distribution is not suitable. (Based on 2σ/3σ rule)

Can someone please explain to me how what the mark scheme shows means that the distribution is not suitable, as I can't figure it out. Also, what is the '2σ/3σ rule'? Thank you very much for your help!




https://en.wikipedia.org/wiki/68%E2%80%9395%E2%80%9399.7_rule

These two things should help explain it for you :h:

After you have read those things...

the question is using the rule to determine that:

from
90 + 2σ = 173, '95%' of visits will take less than 173 minutes. (This is not strictly true, it's actually 97.5% due to the other tail but doesn't matetr for what we're talking about)

And 215 is the '99.7%' where 99.7% (really 99.85, due to the tail) will take less than 215 minutes.

Hence for you to be 95% sure that no one will be there after 10pm, the latest visit start time should be the one given.

To be 99.7% sure, the earlier time of 6:25 is what is needed.

Hence if you put in 8pm then sometimes people's visits will go past 10pm which is not possible...and so..

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