1. Definition of subjective probability. Just Google it (or other search engine). And you might want to know about objective probability too.
Defination from investopedia.com:
A probability derived from an individual's personal judgment about whether a specific outcome is likely to occur. Subjective probabilities contain no formal calculations and only reflect the subject's opinions and past experience.
2.Simple experiment such as picking a number-written paper from a container with replacement. It doesn’t matter if the number generated isn’t random enough since the question only tell us to generate 30 3-digits NUMBERS, not RANDOM numbers
The so-called “REASONABLE probability” is actually the subjective probability (as mention in Q4 “…… to revise subjective probability obtained in step 2.”, and the question doesn’t mention other probability), not the probability obtained from the sample which is the objective probability. So it’s based on your personal judgment, not from recorded observations. For example: based on my personal judgment, the probabilities for 3-different, 2-same and 3-same are 0.60, 0.39 and 0.01 respectively.
The deduction part? Maybe you can say that the 3-different-digits-numbers has a higher chance of occurring, followed by 2-same-digits then 3-same-digits.
**DO NOT mention anything about the theoretical probability (0.72,0.27 and 0.01 respectively).This assignment assumes that you DON’T HAVE ANY KNOWLEDGE ANOUT IT.
This whole assignment is NOT about investigating how "RANDOM" the number generated by using different method. It's about investigating how many number of 3-different, 2-same, and 3-same in the range of 000-999.
3. Random number function. The simplest way is to use the Ran# function in your calculator,multiply it by 1000. Since the max value of Ran# is 0.999 and don’t include 1,max value of 1000Ran# will be 999, not 1000.You can also use the RANDBETWEEN functionin Microsoft Excel. Just type =RANDBETWEEN(0,999) and pull it down and got a whole list of random numbers. And if you have some programming skill you can create some sort of application to call the function (rand() function in PHP,for example).
And if you plan to use the online random number generator for this question, please make sure you know how they generate random number, because not all random number generators use random number function. Know this: some of the websites just use the phprand() function or similar, while some websites like random.org utilize external source such as atmospheric noise, lavalamp or radioactive decay to make the numbers generated more random compared to pseudo-number generator like the simple random number function.
Strongly advise you to use the 1st and 2nd method.
Comment on the answer? Maybe talk about length of interval and thestandard error. And since the confidence intervals for 3-same-digits-numbers contain negative values, and p cannot be a negative value, this indicates that(in my opinion)
1. Confidence interval is not a good approach for3-same-digits-numbers in this case.
2. Assuming it’s normally distributed, you can’t be 90% or 95% symmetrical confidence for 3-same-digits-number in this case.
3. The sample size (n=100) is still too small. (a larger sample size can solve this problem, such as n=1000)
4 (a) If your suggested probability in Q2 falls in both 90% and 95% confidence intervals in Q3, then you conclude that the suggested probability is reasonable enough to be accepted. Or you can make a little change to it so it’s closer to the p ̂ of confidence intervals (e.g. 0.68 to 0.69)
If your suggested probability in Q2 falls in both 95% but not 90% confidence intervals in Q3, then you can also conclude that the suggested probability is reasonable enough to be accepted. (Since the difference in confidence interval between 95% and 90% one is small). Or you can make a little change to it so it’s closer to the p ̂ of confidence intervals (e.g. 0.67 to 0.69)
If your suggested probability in Q2 doesn’t falls in both 90% and 95% confidence intervals in Q3, then you conclude that the suggested probability is not reasonable enough to be accepted. So you can make change to it so it falls in the confidence interval for 95% (or 90%) one. (e.g. 0.60 to 0.67)
(b)The number on the banknote, ISBN of books or bar code ......, And you have to take some measures to ensure that sample obtained is random such as obataining bar code from different product.
If you have some knowledge in programming you can create an application that converts bytes in a computer file which appear to have a random pattern (such as radio noise, white noise image, etc.). And don't just take the value of bytes directly. Perform some processing so that the number generated is random enough. Here's my example
http://www.sniperkitten.tk/generate-random-number-from-radio-noise/ (written in PHP).
(c) Just perform a chi-squared goodness-of-fit tests, where observed frequency Oi is the frequency from the sample (Q4b) and expected frequency Ei is your revised subjective frequency (Q4a).
5(a) Since the question don’t specify how the random number, n should be generated, the easiest ways are still using calculators and Microsoft Excel. However, please note that the Ran# function in the calculator only produce numbers of up
to 3 demicals only (0.000-0.999) so the number generated may not be good enough (but sufficient for practical purpose). Using RANDBETWEEN(0,1) in Excel don’t work as it produce string of 1s and 0s. Use
=RAND() instead which produce number from 1 to 0 (1 exclusive) up to 9 decimals, which is better than the calculator counterpart. Then, suggest a value for the probability, or your subjective probability, for 3-different digits. Again,it's up to you.
(Just FYI, no need to show) Here's the theoretical probability forthis question:
http://www.sniperkitten.tk/theoretical-probability-of-every-number-from-000-999-generated-from-1000sqrt5n/
(b) Just use hypothesis test for large population proportion (Since np>5 and nq>5), one-tailed (since the question only require “more than …”)
发表于 5-7-2013 05:47 AM
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