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S1 - Question on the normal distribution

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! :smile:

Reply 1

Dekota

sweet_gurl

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'?

Let X be the length of the visit, then
X ~ N(90, 41.587²)

If Tara arrives at 8pm, she is expected to leave the club at 8hrs + 1.5hrs = 9:30pm

Basically the model is not suitable because, Tara is expected to leave the club 30 mins before it closes, and she can't stay any longer. This means there cannot not even be a difference of 1 standard deviation in her leaving time, so this normal distribution curve becomes invalid for her arrival-leaving time.

Anyone else got better suggestions? ..

Reply 2

Kernel

I don't get what the markscheme is talking about, but I think it'd be sufficiant to say that the prob of her visit being longer than 2 hrs is 0, so normal distribution is not a suitable distribution.