A surveyor is measuring the height of a cliff known to be about 1000 feet. He assumes his instrument is properly calibrated and that his measurement errors are independent, with mean μ = 0 and variance σ2 = 10. He plans to take n measurements and form the average. Estimate, using (a) Chebyshev’s inequality and (b) the normal approximation, how large n should be if he wants to be 95 percent sure that his average falls within 1 foot of the true value. Now estimate, using (a) and (b), what value should σ2 have if he wants to make only 10 measurements with the same confidence?
To review, Chebyshev’s inequality for a sample mean is:
P(|X - mean| >= k) <= variance/(N * k^2)
Here, we solve for:
.05 <= 10/(N * 1^2)
If he wants to be 95% sure that his average falls within 1 foot of the true value, he should take at least 200 measurements according to Chebyshev’s inequality.
Next, to find how much lower the variance would have to be for him to be this confident with only 10 measurements, we solve for the following equation:
.05 <= variance/(10 ^ 1^2)
If he wants to be equally confident with only 10 measurements, he would need a measurement method with a variance of no more than 0.5 feet according to Chebyshev’s inequality.
We start with variance = 10 and standard deviation = sqrt(10).
So the standard error of the mean, which is equal to sd/sqrt(n), is given by sqrt(10)/sqrt(n).
Next, let’s find the z-score for the 95% confidence interval.
qnorm(.025)## [1] -1.959964We find that the 95% confidence interval is bounded by +/- 1.96 standard deviations.
To be 95% sure that the average falls within 1 foot of the true value, we therefore want 1.96 x the standard error of the mean to be no more than 1.
Solve for the equation 1.96 x sqrt(10)/sqrt(n) = 1.
We get 1.96^2 x 10, which is approximately:
1.96^2 * 10## [1] 38.416We cannot have partial measurements, so round up to 39.
If he wants to be 95% sure that his average falls within 1 foot of the true value, he should take at least 39 measurements according to the normal approximation.
Now, what if he can only take 10 measurements? In that case, he would need a more precise way of measuring, aka his measurement method would need a smaller variance.
If we solve for the equation 1.96 x sd/sqrt(10) <= 1, we get that the standard deviation would need to be no more than sqrt(10)/1.96, or the variance would need to be no more than 10/1.96^2.
This is approximately:
10/(1.96^2)## [1] 2.603082He would need the measuring method to have a variance of at most 2.60 feet to measure with the same confidence using only 10 measurements according to the normal approximation.