Help for coin grading % error, odds & conclusions

[aques]

I need to ask a question that has several parts. It has to do with error percentages, odds & conclusions if any can be given. Where do I post the question. I think the questions & calculation are simple & probably could be answered in a high school math class but it's beyond my mathematical abilities. I'm near 70 years old & I have been away from any connection to math for a long time. I see by some of the questions on this site my questions may be to simple, but I would appreciate your help anyway. Where would I post my questions? It will not be a homework question-lol

 

[torquil]

I'm not a PhysicsForums administrator, but I would suggest to just ask here or in the "Set Theory, Logic, Probability, Statistics" section. Since you have created this thread you might as well ask here in this thread.

I don't think any question is too simple for this place.

 

[Greg Bernhardt]

Welcome aques! If you are unsure you can post right here in "General Math" and one of our fine mentors will move it if needed.

 

[aques]

OK-I will post here. First a little background. I’m not sure how many of you are familiar with the grading of coins. There are only a few respected companies that grade coins. PCGS is one of the top 2 & in my opinion the best. Most un-circulated coins are graded mint state (MS) 60-70 & proofs (PF) 60-70. A MS/PF70 being the highest grade (Supposedly a perfect coin). There can be and usually is a huge difference in the price from an MS/PF69 to a MS/PF70. You also have the option on these coins of having a normal label or a first strike label All or nearly all qualified for FS. All had to be sent to the grading co. un-opened as sent from the US Mint. If you need pictures, I will post them. You must state your preference before they will grade your coin. The first strike label is an additional $18 & the cost to the company for the labels is probably the same for both labels. The US Mint sold a 25th Anniversary ASE (American Silver Eagle) set. It included:
*one proof coin from the United States Mint at West Point
* one uncirculated coin from the United States Mint at West Point
* one uncirculated coin from the United States Mint at San Francisco
* one reverse proof coin (frosted background with polished, mirror-like design elements/foreground) from the United States Mint at Philadelphia
* one bullion coin
That is a little of the background & if more info is needed just ask. I’m interested in only MS/PF70 graded coins of all 5 coins above. These are the grades in MS/PF70 grades for the 5 coins in question. I'm not sure why my table did not come out the way it shows when I wrote it & may be hard to read. Just remember the numbers after the type coin are non first strike & the % are also in that order. Sorry!

MS/PF 70’s

Non- First Strike First strike
Bullion coin 202 MS70 out of 616 coins 5,628 MS70 out of 10,700 coins c for 32.8% for 52.6%

S mint Un-cir. 153 MS70 out of 616 coins 5,786 MS70 out of 11307 coins
for 24.8% for 51.2%

W mint Un-circ 214 MS70 out of 606 coins 5,835 MS70 out of 10,660 coins
for 35.3% for 51.7%

W mint Proof 219 PF70 out of 608 coins 5,766 PF70 out of 10,699 coins
for 36% for 53.9%

P mint revpf 223 PF70 out of 594 coins 5,960 PF70 out of 11,314 coins
for 37.5% for 52.7%

Now for some questions—You notice more coins are graded with the first strike label. What would be % of error for the smaller # of non first strike coins be compared to the larger # of first strike coins. In other words is the non first strike coins to small a number to get a comparison between the two? It seems to me there are only 2 choices MS70 or a lower grade & 600 coins would be enough of a sample. I think that the more coins graded would give better results, but I’m not sure it would account for the wide % differences. I do not know how to calculate this--help
The second question is there a way to figure the odds that all 5 coins would have a substantial % difference between the first strike & the non first strike. The first strike having a much larger % graded @ the MS/PF 70 grade.
A third question I have is---- Would it make you think that something is influencing the grade that the coin receives. Since they will not grade the coins until you make the decision on the type of label you prefer (an extra $18/coin), would that raise a red flag? The MS/PF 70’s also sell for a substantial premium - much more than the extra $18 grading fee per coin.
One more question----If there are any equations used, could you please give them to me? Any help would be greatly appreciated. If you need any more information, please ask-----Thank you

 

[aques]

maybe this will show it better

 

[aques]

I'm a little disappointed. Is my question to simple, not answerable or is more info needed. I would be happy to give more info if it is needed--just tell what you need

 

[pbuk]

It's not entirely clear to me what your question means, let's see if I understand it, taking the bullion coins as an example.

5,628 out of 10,700 first strike coins (approximately 53%) were graded MS/PF70. Only 202 out of 616 non-first strike coins (approximately 33%) were graded MS/PF70. How likely is it that non-first strike coints are as 'good' as first strike coins?

Well, the first 2 numbers indicate that the probability of a first strike coin being graded MS/PF70 is 0.5260. We make what is called a 'null hypothesis' that the probability p of a non-first strike coin being graded MS/PF70 is also 0.5260. In statistics there is something called the normal distribution, known informally as a bell curve. The bell can be tall and thin or widely spread out: this is characterised by a parameter called standard deviation.

If you do many trials taking n coins each time it can be shown that the number that are graded MS/PF70 will follow a normal distribution with mean np and standard deviation given by √(np(1-p). We can then use statistical tables, or the formulae built into a spreadsheet or a statistics program to 'test the null hypothesis'.

So in this case n=616 and p=0.5260. The probability that out of 616 coins, 202 or fewer are graded MS/PF70 is given by the cumulative normal distribution function with mean 616 x 0.5260 = 324 and standard deviation √(616 x 0.5260 x (1 - 0.5260)) = 12.39. I used Excel (the formula is NORM.DIST(x,Mean,StandardDeviation,TRUE) where x is the number graded MS/PF70) to get the probability of 3.6 x 10^-23. This is unimaginably small, so it is inconceivable that the null hypothesis is true. We therefore concude that non-first strike bullion coins are not as likely to be graded MS/PF70 as first strike bullion coins.

 

[aques]

Are the sample sizes large enough in the non first strike coins to come to that conclusion? Thanks Mr. Anchovy for the info. & thanks for taking the time to answer my questions---Ike

 

[pbuk]

Providing the sample is unbiased, it is plenty big enough. Indeed with a sample of only 60 bullion coins with a 0.5260 probability of being graded MS/PF70 we would expect to get 20 or fewer such gradings only once in 500 samples.

(note that when using a smaller sample size a continuity adjustment becomes important: the adjusted formula is NORM.DIST(x+0.5,Mean,StandardDeviation,TRUE)).

 

[aques]

Thank you very much. I think you have answered my questions!