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Expert: - 6/20/2009


Question
QUESTION: A lorry load of potatoes has, on average, 1 rotten potato in 6. A greengrocer tests a random sample of 100 potatoes & decides to turn away the lorry if she finds more than 18 rotten potatoes in the sample. Find the probability that she accepts the consignment.

ANSWER: Use n = 100 and p = 1/6, so 1 - p = 5/6.

Approximate this with a normal distribution.

The mean is np and the variance is np(1-p).

The standard deviation is the squareroot of the variance.

For the limits, it has been said to find the P(n<=18),
when approximating with a normal, 18.5 should be used,
since anything less rounds to 18.

The averge would be 100/6 = 50/3 = 16 2/3.
The variance would be 100(1/6)(5/6) = 125/9.
This means the standard deviation would be 5*sqrt(5)/3.

Using this find out how many standard deviations 18.5 is over 16 2/3.

18.5 - 16 2/3 = 18 3/6 - 16 4/6 = 1 5/6, or around 1.83333...,
so the standard deviation is 1.27...

So what you need is the probability that the normal distribution is above 1.44 standard deviations away from the average.


---------- FOLLOW-UP ----------

QUESTION:  Why p(x<=18) ? Why not p(x>18) since the question said more than 18 and the question didnt said it is inclusive, why put = sign? Why p=1/6? How do you know n=100? I have difficulty in determining which is n and which is p. Can you explain? Cheers

Answer
I was 1/6 since one of of 6 potatoes is bad.  If she finds 18 or less, the lot is accepted.

The P(x<=18) is the chance of accecptance.  A normal probability distribution would use P(x<=18.5) for the chance of acceptance.
It would be P(x>=19.5) for the chance of rejection.

We use n=100 because n is the sample size.

Note the first letter of each.
The letter 'n' is for number since 'n' is the first letter.
The letter 'p' is for probability since 'p' is the first letter.

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