Question 7, Page 355

A surveying instrument makes an error of −2, −1, 0, 1, or 2 feet with equal probabilities when measuring the height of a 200-foot tower.

1. Find the expected value and the variance for the height obtained using this instrument once.

2. Estimate the probability that in 18 independent measurements of this tower, theaverageofthemeasurementsisbetween199and201, inclusive

Answer

Equal Probabilities for -2, -1, 0, 1 and 2 feet for error.

(a). expected value

av <- (-2+-1+0+-2+2)/5 # average error
m <- 200+av
m

## [1] 199.4

2. variance

v_1 <- (-2)^2
v_2 <- (-1)^2
v_3 <- (0)^2
v_4 <- (1)^2
v_5 <- (2)^2

avar <- (v_1 + v_2 + v_3 + v_4 +v_5)/5
avar

## [1] 2

2. 18 independent measurements. n =18 \(x^{-}=199 - 201\)

n <- 18
p_1 <- (199-200)/sqrt(n)
p_1

## [1] -0.2357023

p_2 <- (201-200)/sqrt(n)
p_2

## [1] 0.2357023