# Using normal distribution

#### spot

##### New member
Hi, I've got this question

Using the standard normal distribution X ~ N(0,1), answer the following question:

A pet shop owner is selling rabbits. The weight of the rabbits is normally distributed with mean 1000g and a standard deviation 100g. Rabbits between 900g and 1100g are sold as ‘Standard Rabbits’ the rest are sold as ‘Special Rabbits’. The pet shop owner wants to raise equal money from the sale of Special and Standard Rabbits. If Standard Rabbits are priced at £5.00, what should be the price of a Special Rabbit?

My attempt:

31.6% of rabbits are special, 68.4% are normal
68.4 / 31.6 = 2.16
2.16 * 5 = 10.8
£10.80

I'd be grateful if someone could help out. Have I got this right?
Thanks

#### tkhunny

##### Moderator
Staff member
Hi, I've got this question

Using the standard normal distribution X ~ N(0,1), answer the following question:

View attachment 10766

A pet shop owner is selling rabbits. The weight of the rabbits is normally distributed with mean 1000g and a standard deviation 100g. Rabbits between 900g and 1100g are sold as ‘Standard Rabbits’ the rest are sold as ‘Special Rabbits’. The pet shop owner wants to raise equal money from the sale of Special and Standard Rabbits. If Standard Rabbits are priced at £5.00, what should be the price of a Special Rabbit?

My attempt:

31.6% of rabbits are special, 68.4% are normal
68.4 / 31.6 = 2.16
2.16 * 5 = 10.8
£10.80

I'd be grateful if someone could help out. Have I got this right?
Thanks
1) How did you get 68.4%
2) Estimation Exercise: There are about half as many "special", so you'll need about twice the price. This should make your result seem reasonable.

#### spot

##### New member
Just did the sum again (15 + 19.1) * 2 = 68.2%
Looks like I made a silly error!

#### tkhunny

##### Moderator
Staff member
Just did the sum again (15 + 19.1) * 2 = 68.2%
Looks like I made a silly error!
This illustrates one way you can check your work. You can find 68.2 by adding the four pieces, like you just demonstrated, or you can add up all the other pieces and subtract that from unity (1). Should be the same.

#### spot

##### New member
Sounds good
So my method divides those two percentages. The problem is, I can't figure out why I do the division. It just seems like instinct. I don't really like these kinds of methods (it's essentially guessing what to do). Would you know an alternative method to solve this (or maybe have an explanation for this one)?
Thanks

#### tkhunny

##### Moderator
Staff member
Sounds good
So my method divides those two percentages. The problem is, I can't figure out why I do the division. It just seems like instinct. I don't really like these kinds of methods (it's essentially guessing what to do). Would you know an alternative method to solve this (or maybe have an explanation for this one)?
Thanks
There a vast difference between simply "guessing" and knowing where to guess. You seem to be in the latter method. This is good.

#### Dr.Peterson

##### Elite Member
A pet shop owner is selling rabbits. The weight of the rabbits is normally distributed with mean 1000g and a standard deviation 100g. Rabbits between 900g and 1100g are sold as ‘Standard Rabbits’ the rest are sold as ‘Special Rabbits’. The pet shop owner wants to raise equal money from the sale of Special and Standard Rabbits. If Standard Rabbits are priced at £5.00, what should be the price of a Special Rabbit?

My attempt:

31.6% of rabbits are special, 68.4% are normal
68.4 / 31.6 = 2.16
2.16 * 5 = 10.8
£10.80
So my method divides those two percentages. The problem is, I can't figure out why I do the division. It just seems like instinct. I don't really like these kinds of methods (it's essentially guessing what to do). Would you know an alternative method to solve this (or maybe have an explanation for this one)?
I appreciate your concern: you want to have a reason for what you do, to make sure you aren't following the wrong instinct.

You could use algebra; I'd think in terms of proportions, or maybe just units. I'll just talk through it and see what seems natural to me.

You have 31.6 special rabbits and 68.4 normal. You want the total cost of each group to be equal: 31.6*cost per special = 68.4*cost per normal. So the costs are inversely proportional to the number, and you divide; algebraically, 31.6*cost per special = 68.4*5.00, so cost per special = 68.4*5.00/31.6.

Any of those ideas may have been going through your mind.

Another way you can convince yourself you're right is to do exactly what tkhunny suggested, checking that it makes sense. With about twice as many normal, they should cost half as much to get the same total. If they cost twice as much, the total would come to 4 times as much.

#### spot

##### New member
Hooray, algebra saves the day!
It's amazing what a single line of algebra can do.
That's put my mind to rest, I get it now. Thanks!

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