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.

• 0.975
• 0.98
• 0.983
• 0.99
• 0.997

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.