Magnetic surveying is one technique used by archaeologists to determine anomalie

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.

0 x < 10

10 x < 20

20 x < 30

30 x < 40

40 x

Suppose a "hot spot" is a site with a reading of 40 or higher.

a.) Suppose you decide to take readings at random until you get your first "hot spot." Let n be a random variable representing the trial on which you get your first "hot spot." Use the geometric probability distribution to write out a formula for P(n).

P(n) = _____

   

b.) What is the probability that you will need more than four readings to find the first "hot spot"? Compute P(n > 4).

(Round your answer to three decimal places.)

2.) 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.

0 x < 10

10 x < 20

20 x < 30

30 x < 40

40 x

Suppose you are working in a "warm" region in which all MS readings are 20 or higher. In this same region, what is the probability that you will find a "hot spot" in which the readings are 40 or higher? Use conditional probability to estimate

P(40 x | 20 x).

(Round your answer to three decimal places.)

P(40 x | 20 x)

=

Magnetic Susceptibility Readings,
centimeter-gram-second 106 (cmg 106) Comment Magnetic
Susceptibility Number of
Readings Estimated
Probability "cool"

0 x < 10

36 36/120 = 0.30 "neutral"

10 x < 20

48 48/120 = 0.40 "warm"

20 x < 30

24 24/120 = 0.20 "very interesting"

30 x < 40

6 6/120 = 0.05 "hot spot"

40 x

6 6/120 = 0.05

Explanation / Answer

a.) Suppose you decide to take readings at random until you get your first "hot spot." Let n be a random variable representing the trial on which you get your first "hot spot." Use the geometric probability distribution to write out a formula for P(n).

P(n) = (1 - 0.4)n-1 (0.4) = 0.6n-1 0.4

(b) What is the probability that you will need more than four readings to find the first "hot spot"? Compute P(n > 4).

P(n >4) = 1 - P(n <=4) = 1 - [1 - (1 - 0.4)4] = 0.64 = 0.1296

Question 2

Suppose you are working in a "warm" region in which all MS readings are 20 or higher. In this same region, what is the probability that you will find a "hot spot" in which the readings are 40 or higher? Use conditional probability to estimate

P(40 x | 20 x) = P(40 x)/ P(20 x) = P(40 x)/ [P(20 x <30) + P(30 x <40) + P(40 x)] =

= 0.05/ (0.15 + 0.10 + 0.05) = 1/6 = 0.167

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