If the actual proportion = 0.5, we can find the variance by the formula p(1-p):

Variance = 0.5*0.5 = 0.25

So the standard deviation for a single sample will be:

SD = sqrt(variance) = 0.5

Now, the question confusingly calls the estimate variability the standard deviation, but we more normally refer to this as the Standard Error of the Mean (SEM), the variability around an estimated (sample) mean. We find this by dividing the standard deviation by the square root of n, our number of samples:

SEM = SD/sqrt(n)

= 0.5/10

= 0.05

(See how, as you add more samples - n increases - you get a smaller error around your estimated mean? If you poll 5 people you won't be too accurate, but if you poll 1000 you'll get much more accuracy. The single sample standard deviation doesn't change though! Each individual will still be 0.5 away from the mean, i.e. they'll be 0 or 1 for liking sushi!)

Mar 19th, 2015

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