A surveying instrument makes an error of [math]-2[/math], [math]-1[/math], 0, 1, or 2 feet with equal probabilities when measuring the height of a 200-foot tower. Estimate the probability that in 18 independent measurements of this tower, the average of the measurements is between 199 and 201, inclusive.

References

Doyle, Peter G. (2006). "Grinstead and Snell's Introduction to Probability" (PDF). Retrieved June 6, 2024.

Solution: E

Let [math]\sigma^2 [/math] equals the variance of a single measurement error. A quick calculation gives [math]\sigma^2 = 2[/math]. The average measurement is the sum of the measurements divided by 18, and is approximately normally distributed with mean 200 and variance [math]\sigma^2/18 = 1/9[/math], and the approximate probability that the average measurement is between 199 and 201 equals

[[math]] P(Z \in \pm 3 ) = 0.9973 [[/math]]

where [math]Z[/math] is a standard normal variable.