Wikipedia:Reference desk/Archives/Mathematics/2006 December 26

Welcome to the Wikipedia Mathematics Reference Desk Archives
The page you are currently viewing is an archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages.

December 26

Cartesian geometry

Can I get some help solving this problem-

What are the number of integer values of m,for which the x coordinate of the point of intersection of the lines 3x+4y=90 and y=mx+1 is also an integer?

Thank you. —Preceding unsigned comment added by 59.93.67.55 (talk • contribs)

First, solve the simultaneous equations

$3x+4y=90$

$y=mx+1$

(treating m as a constant) to find the value of x in terms of m. You will find that x is the result of dividing an integer numerator by an expression in m. For x to be an integer, the expression in m in the denominator of this fraction must be a factor of the numerator. The numerator only has a finite number of factors - try each one, and solve for m to see if m is an integer (don't forget to try the negative factors of the numerator too). Gandalf61 10:35, 26 December 2006 (UTC)

Business Statistics

Once upon a time, I learned how to do this in college. But now I've forgotten. Let's say I have a store that is trying to track the sale of widgets and determine the appeal of of the widget to certain segments of the population. For example, I have determined that the store has made sales to 350 customers who have red hair. Of those customers, 175 (50%) of them have purchased widget "A". I have determined that there are 12,000 people with red hair who live within the store's market. I would like to say that "Based upon my experience, I can say with N% certainty that 50% of all customers with red hair would buy widget "A". " In other words, I need to determine whether, for a given statistic, I have a large enough sample size in order to draw a conclusion about the population as a whole, or to assign a value to the degree of confidence I should hold in my sample size. Any help would be appreciated. Neil916 (Talk) 21:34, 26 December 2006 (UTC)

To phrase this in statistical terminology: You have a population (in this case: all the people with red hair who might come into your store), a sample (all the events of members actually entering the store), an assumption of an independent identically distributed random variable for the members of the population (in this case a binary variable: to buy or not to buy, that is the question) determined by one or more population parameters (in this case 1 parameter, the probability of buying), and an observation of the values of that random variable on the events corresponding to the events of the sample. Let us ignore the issue of customers entering the store multiple times. Because of the identical distribution, the information in the sample can be represented with two aggregate values: the size N of the sample, and the number B of buyers. So B can, in principle, be any value from 0 to N. Let p denote the parameter we are trying to estimate.

If our statistical model is correct, then for any given value of p we can calculate the probability that (given the value of N) we would observe the value of B that we did. For example, if N = 350 and p = 0.99, then it is very hard to explain if we observed B = 2. We would have expected something like B = 346. An observation B = 339 might perhaps still be believable, but not B = 2. More likely, we are mistaken in the belief that p = 0.99. Given any putative value for p, we can say: with a probability of (say) 95%, the value of B for a random sample with size N will be in this interval: ..., where the actual interval depends on p. There may be several intervals with that property, but we can take the smallest. The value of 95% is called the confidence level, and the interval is the confidence interval. Now comes a trick that is not wholly justifiable, but widely applied. Given a set confidence level, which you pick in advance, where the height depends on how certain you want to be in what you say, you can define that a value of p "allows" a value of B if B is in the confidence interval corresponding to p. We have B and want to estimate p. Well, take all possible values of p that allow the observed value of B, and say: the "true" value of p of our population must be somewhere among this set.

Now all we have to do is figure out how to determine these confidence intervals. I can already tell that the aggregate value B follows a binomial distribution. Let me know if this helps you on the way.  --Lambiam^Talk 23:25, 26 December 2006 (UTC)

PI question

The sequence "314" appears in the first 3 terms of the sequence of PI (obviously). My question is how many digits has to pass before it reappears again? 202.168.50.40 22:01, 26 December 2006 (UTC)

According to these digits of pi, 314 appears as the 228-230th digits. There was a lot of counting involved, so I probably made some stupid mistake somewhere. PullToOpen^Talk 22:09, 26 December 2006 (UTC)

Well, I ran a search on my 10,000 digits of pi and found the first "314" at 2120 decimal places. Also, your counting seems to be wrong. The occurrence is clearly after the 200th digit. Oh, and remember the number is written as pi, not PI (that makes it look like an acronym, which it isn't). — Kieff 22:34, 26 December 2006 (UTC)

The first 100 occurrences are: 1: 0, 2: 2120, 3: 2538, 4: 2762, 5: 3496, 6: 3903, 7: 4246, 8: 5110, 9: 5233, 10: 5743, 11: 6178, 12: 7575, 13: 8240, 14: 8244, 15: 8561, 16: 8921, 17: 9452, 18: 12085, 19: 12651, 20: 12916, 21: 14649, 22: 17195, 23: 18512, 24: 19466, 25: 20578, 26: 24213, 27: 24369, 28: 24598, 29: 26120, 30: 26529, 31: 26612, 32: 29078, 33: 30116, 34: 31244, 35: 31316, 36: 33685, 37: 34408, 38: 35929, 39: 38732, 40: 38808, 41: 40858, 42: 41375, 43: 42125, 44: 42358, 45: 43056, 46: 43256, 47: 43991, 48: 45902, 49: 45953, 50: 50617, 51: 50681, 52: 50827, 53: 51443, 54: 51531, 55: 52271, 56: 53276, 57: 56026, 58: 57217, 59: 57670, 60: 58546, 61: 60954, 62: 61226, 63: 66841, 64: 68918, 65: 68942, 66: 68952, 67: 69526, 68: 71000, 69: 72712, 70: 73401, 71: 76011, 72: 78569, 73: 79595, 74: 79935, 75: 80125, 76: 80441, 77: 81265, 78: 82352, 79: 83096, 80: 86411, 81: 86659, 82: 87632, 83: 88008, 84: 90893, 85: 91242, 86: 91354, 87: 93728, 88: 94800, 89: 95961, 90: 96201, 91: 97339, 92: 97852, 93: 98107, 94: 98348, 95: 99914, 96: 106547, 97: 107402, 98: 108446, 99: 109579, 100: 110068. The first re-occurrence of "314159" is at the 176451st decimal place.  --Lambiam^Talk 22:47, 26 December 2006 (UTC)