Question

Magnetic surveying is one technique used by might not) indicate an important archaeological discovery. Let x be a random research location A random sample of 120 sites gave the readings shown in the tabile belovw s archaeologists to determine anomalies anising from varlations n magnetic suscegtibility Unusual changes in megnetic suscepbuly ght or varlable that represents a magnetic susceptiblity (MS) reading for a randomly chosen ste st an rchelogical centimeter-gram-second-10-(emg 10-) Magnetic Number of Estimated 36 48 12 18 6/120-0.30 48/120-。40 2/120-0.10 18/120-0 15 6/120-0.05 105x420 warm Thot spet i spots* in the 100 readings region. We have a binemial setting where success is a "hot spot" In this case, the probabilty of success is p. pr 40 s x) The archaeologist takes n-100 magnetic susceptbiny readings nuo nen, nexperedaeaut, benner al random vanabterepresenti,the number o"o (a) We want to approximate the binemial random variable r by a Poisson distribution. Is this appropriate? What requirements must be satisfied before we can do this? Do you think these requirements are satisfied in this case? Explain O No, it is not appropriate since at least one of the criteria np >5 and nq> 5 are not satisfied O Yes, it is appropriate srce the criter", z 100 nnd np < 10 are are both satisfied. OYes, iR is appropriate since the criteria np 5 and nq > 5 are bath setisfied O Yes, t is appropriate since n 30 No, it is not appropriate since at least one of the criteria n 2 100 or np < 10 are not satisfied What is the value of A? (b) Use the Poisson distribution to approximate the probability that the archaeologists well find four or fewer hot spots. int Use Table 4 of Appends t. (Round your anower to four decimal places.

Answer 1

r is a random variable representing the number of hot spots in 100 readings.

r has a Binomial distribution with parameters, number of trials n=100 and success probability p=0.05

We can use Poisson distribution to approximate a Binomial if $n\ge 100$ and p is low such that $np<10$

Here n=100 and $np=100\times 0.05=5$ . Hence we can use the Poisson approximation.

ans: Yes, it is appropriate since the criteria $n\ge 100$ and $np<10$ are both satisfied

The expected value of a Binomial distribution is

$E(R)=np=100\times 0.05=5$

We use a Poisson approximation with parameter

$\lambda=np=100\times 0.05=5$

ans: $\lambda=5$

The Poisson probability that r hot spots are found is

$P(r)=\frac{\lambda^r e^{-\lambda}}{r!}=\frac{5^r e^{-5}}{r!},\quad r=0,1,...$

b) The probability that archaeologists will find four or fewer hot spots is

$\begin{align*} P(r\le 4)&=P(r=0)&plus;P(r=1)&plus;P(r=2)&plus;P(r=3)&plus;P(r=4)\\ &\text{using table 4 of appendix II (Poisson probabilities), for }\lambda=5\text{ and }x=0,1,2,3,4\\ &=0.0067&plus; 0.0337&plus; 0.0842&plus; 0.1404&plus; 0.1755\\ &=0.4405 \end{align*}$

ans: The probability that archaeologists will find four or fewer hot spots is 0.4405

c) The probability that archaeologists will find more than seven hot spots is

$\begin{align*} P(r>7)&=1-P(r\le 7)\\ &=1-(P(r=0)&plus;P(r=1)&plus;...&plus;P(r=7))\\ &\text{using table 4 of appendix II (Poisson probabilities), for }\lambda=5\text{ and }x=0,1,2,...,7\\ &=1-(0.0067&plus; 0.0337&plus; 0.0842&plus; 0.1404&plus; 0.1755&plus; 0.1755&plus; 0.1462&plus; 0.1044)\\ &=1-0.8666\\ &=0.1334 \end{align*}$

ans: The probability that archaeologists will find more than seven hot spots is 0.1334