Re: margin of error

[dennis roberts]

since i followed the model of m and m ... and, you seem to think their wording is ok ... then, by inference ... so is mine

what is puzzling about your original disagreement with my handout note is the following:

Where it says,
"how large of a SAMPLE would I need,
with 95% confidence, to produce an interval ...." - >>>> your words >>>>> it ought to say,
"... with 50% confidence, ..."

you claim that the method i illustrated essentially was working with a fifty (50%) confidence interval (level) ... where did you come up with that?

let's assume for the moment ... that we are taking samples from a population where the mean is 100 and the sigma is 16 ... like iq scores

i generated 1000 samples of n=16 ... found the means ... then, using a t value based on about 15 df ... when sample mean +/- 2.1315 units of error (16/sqrt 16 = 4) ... and found that about 96 % of these intervals captured 100 ... ie, about 96% of the intervals USING A MARGIN OF ERROR OF 2.1315 * 4 = 8.5 ... CAPTURED the mu value ... so, the mu value was within + OR - 8.5 about 96% of the time ...

NOT 50%

what if you wanted the margin of error to be about 1/2 of that ... or instead of 2.1315 * 4 ... it would be about t * 2 ... and even though the t value would have to be a bit smaller ... i still used "as an approximation" a t value of 2.1315 ... as my handout clearly stated ...

so, i generated 1000 new samples from the same population where n = 64 ... and, with the standard error about 2 (16/ sqrt 64) rather than 4 ... which makes the margin of error about 1/2 of the original size ... i went this margin of error above and below the mean ... and then counted up how many times 100 or mu was inside this interval that is now ABOUT HALF THE WIDTH of the previous one when n=16

i found 97% of the intervals included mu ... thus, 97% of the time ... the intervals were within about 1/2 of the original margin of error from mu ... ie the same basic level of "confidence" now applies to being CLOSER to mu ...

of course, since i just used t of 2.1315 and, we know it would need to be somewhat smaller ... the real distance around the sample means would have been a tad narrow and not have captured the mu value quite as much as 97% ... so, it would have been closer to 96% or 95%

NOT 50%

thus, in both of the examples i used in my handout and, the two above ... i claim that with 95% confidence ... the intervals around the sample means are WITHIN THE SPECIFIED MARGINS of error ... of course, a small number of times (about 5%) the interval misses it by more than m

the formula used in the handout was to estimate the n you would need to change the margin of error to some specified amount and, what i have shown above and what the formula for n would show is that if you want m to be about 1/2 the size ... you need an n that is about 4 times larger ... and maintain the 95% confidence level

where are you coming up with the 50% level???

.
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