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Physics, Astronomy & Cosmology / Re: Homework help. Statistics for Physics.
« on: 21/01/2022 17:22:06 »
Hi again.
I might be talking to myself here, so I'll make this the last post unless someone else replies.
What do you think that section (g) is actually asking you to calculate? To check you've understood the question and calcuated the right thing, let's just ask if you could usefully advise a group of researchers that they should take N individual measurments if they want to "detect" the change in mass due to velocity. To phrase this another way, how you will you interpret what it means for the change in mass to be detected? Do any of the following apply?
(a) If a group of researchers take N individual measurments and average these, they will be able to determine that the relativistic mass (of protons at 0.2c) is not the same as the rest mass at the 95% confidence level. If they took less than N individual measurements then they won't be able to determine this.
(b) The results that a group of researchers obtain are random. They might be able to reject the null hypotesis (at the 95% level) that the relativistic mass is the same as the rest mass, they might not. The more individual measurments they take and combine to generate an average, the better their chances of being able to reject that hypothesis. Since everything is random they might never be able to reject the null hypothesis even if N is a massive number. However, provided they take N individual measurements and consider the mean value they obtain and the distribution of that mean in the usual way, the probability of them being able to reject that null hypothesis is 95%.
(c) 95% of the time, the mean of N individual measurements will be greater than the rest mass. The researchers should just accept this as an indication that the masses are different. They do not need to do any statistical analysis of their own, they should just go home satisified that the change in mass has been "detected".
(d) Taking N individual measurments and averaging these, the true underlying relativistic mass can be estimated with error bars as usual. We know the observed mean of N individual measurments will be Normally distributed and the standard error of the mean is easily calculated. So 95% of the time the true underlying relativistic mass will be in the range specified by (the observed mean) +/- 2 standard errors of the mean. While 5% of the time it isn't. The value of N doesn't influence these chances at all. The only thing N does is to reduce the standard error of the mean, so that the range of values reported for the relativistic mass is smaller.
(e) None of the above.
I might be talking to myself here, so I'll make this the last post unless someone else replies.
What do you think that section (g) is actually asking you to calculate? To check you've understood the question and calcuated the right thing, let's just ask if you could usefully advise a group of researchers that they should take N individual measurments if they want to "detect" the change in mass due to velocity. To phrase this another way, how you will you interpret what it means for the change in mass to be detected? Do any of the following apply?
(a) If a group of researchers take N individual measurments and average these, they will be able to determine that the relativistic mass (of protons at 0.2c) is not the same as the rest mass at the 95% confidence level. If they took less than N individual measurements then they won't be able to determine this.
(b) The results that a group of researchers obtain are random. They might be able to reject the null hypotesis (at the 95% level) that the relativistic mass is the same as the rest mass, they might not. The more individual measurments they take and combine to generate an average, the better their chances of being able to reject that hypothesis. Since everything is random they might never be able to reject the null hypothesis even if N is a massive number. However, provided they take N individual measurements and consider the mean value they obtain and the distribution of that mean in the usual way, the probability of them being able to reject that null hypothesis is 95%.
(c) 95% of the time, the mean of N individual measurements will be greater than the rest mass. The researchers should just accept this as an indication that the masses are different. They do not need to do any statistical analysis of their own, they should just go home satisified that the change in mass has been "detected".
(d) Taking N individual measurments and averaging these, the true underlying relativistic mass can be estimated with error bars as usual. We know the observed mean of N individual measurments will be Normally distributed and the standard error of the mean is easily calculated. So 95% of the time the true underlying relativistic mass will be in the range specified by (the observed mean) +/- 2 standard errors of the mean. While 5% of the time it isn't. The value of N doesn't influence these chances at all. The only thing N does is to reduce the standard error of the mean, so that the range of values reported for the relativistic mass is smaller.
(e) None of the above.