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I'm curious if there's a good way to estimate the probability that the true mean of a population falls closer to the observed sample mean than the tail ends of the margin of error. That is, I believe that there is unequal probability that the true mean may be greater or less than (to the right or left) of the sample mean.

But, if a survey is conducted and there is a large margin of error, how can we estimate whether it's likely that the true me is closer to the observed sample mean than at the tail end of the margin of error. That is, if the margin of error is -/+10 points, how could you calculate that there's a say 50% probability that the mean is probability -/+ 5points from the observed mean.

Ideally, I'd love a way to calculate this in R, if anyone is also familiar with that programming language.

Thank you and please let me know if i can make this question any clearer

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If you want to calculate the probability a parameter (the mean in this case) falls within a certain distance from a point estimate for that parameter (observed mean) you are basically calculating the confidence level for a specific confidence interval.

The margin of error (ME) is calculated from the critical statistic and the samples standard error (SE):

$ME = critical\_statistic * SE$

Therefore, you can calculate the critical statistic based on the desired margin:

$critical\_statistic = \left(\frac{ME}{SE}\right)$

You can then obtain the p-value of that critical statistic using tables or GraphpPad's p-value calculator. The actual statistic used will depend on the characteristics of your survey (usually a t-statistic).

This p-value would be the probability that the parameter falls outside the desired margin of error, so the confidence level would be $1-p$

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  • $\begingroup$ I suppose what I mean is assume the survey is already completed and there's a given margin of error. How do you figure the probabilities within that band? $\endgroup$ – user3368667 Aug 19 '15 at 5:43
  • $\begingroup$ How did you calculate the margin of error? $\endgroup$ – twalbaum Aug 19 '15 at 5:44
  • $\begingroup$ @user3368667 What I mean is, the margin of error depends on the confidence level you decide (usually 95%). If you show me how you are obtaining that margin of error in R, I can help you with calculating what you want. $\endgroup$ – twalbaum Aug 19 '15 at 6:01

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