Question

Magnetic surveying is one technique used by archaeologists to determine anomalies arising from variations in magnetic susceptibility. Unusual changes in magnetic susceptibility might (or might not) indicate an important archaeological discovery. Let x be a random variable that represents a magnetic susceptibility (MS) reading for a randomly chosen site at an archaeological research location. A random sample of 120 sites gave the readings shown in the table below.

Magnetic Susceptibility Readings, centimeter-gram-second ✕ 10−6 (cmg ✕ 10−6)
Comment	Magnetic Susceptibility	Number of Readings	Estimated Probability
"cool"	0 ≤ x < 10	30	30/120 = 0.25
"neutral"	10 ≤ x < 20	54	54/120 = 0.45
"warm"	20 ≤ x < 30	18	18/120 = 0.15
"very interesting"	30 ≤ x < 40	12	12/120 = 0.10
"hot spot"	40 ≤ x	6	6/120 = 0.05

For a site chosen at random, estimate the following probabilities.

(a)

P(0 ≤ x < 30)

(b)

P(10 ≤ x < 40)

(c)

P(x < 20)

(d)

P(x ≥ 20)

(e)

P(30 ≤ x)

(f)

P(x not less than 10)

(g)

P(0 ≤ x < 10 or 40 ≤ x)

(h)

P(40 ≤ x and 20 ≤ x)

Answer 1

a) P(0 x < 30) = 0.25 + 0.45 + 0.15

= 0.85

b) P(10 x < 40) =0.45 + 0.15 + 0.10

= 0.70

c) P(x < 20) = 0.25 + 0.45

= 0.70

d) P(x 20) = 1 - P(x < 20)

= 1 - 0.70

= 0.30

e) P(30 x) = P(x < 30)

= 0.85

f) P(x not less than 10) = 1 - P(x < 10)

= 1 - 0.25

= 0.75

g) P(0 x < 10 or 40 x) = P(0 x < 10) + P(x > 40)

= 0.25 + 0.05

= 0.30

h) P(40 x and 20 x) = P(x > 40)

= 0.05