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Chapter 8: Q92E (page 452)

Intrusion detection systems. The Journal of Researchof the National Institute of Standards and Technology (November鈥揇ecember 2003) published a study of a doubleintrusion detection system with independent systems. Ifthere is an intruder, system A sounds an alarm with probability.9, and system B sounds an alarm with probability.95. If there is no intruder, system A sounds an alarm withprobability .2, and system B sounds an alarm with probability.1. Now assume that the probability of an intruderis .4. Also assume that under a given condition (intruderor not), systems A and B operate independently. If bothsystems sound an alarm, what is the probability that anintruder is detected?

Short Answer

Expert verified

The probability that anintruder is detected is 0.9661.

Step by step solution

01

Important formula

The formula for probability isP=favourableoutcomestotaloutcomes.

The Baye鈥檚 formula is

P(BiA)=P(Bi鈱�A)P(A)=P(Bi)P(ABi)P(B1)P(ABi)+P(B2)P(AB2)+...+P(Bk)P(ABk)

02

The probability that anintruder is detected

Let,

A=intruder

B=system A sounds alarm

C=system B sounds alarm

PA=0.4PB\A=0.9PC\A=0.95PB\Ac=0.2PC\Ac=0.1

The systems are independent.

PB鈱�C\A=0.9脳0.95=0.855

PB鈱�A\Ac=0.2脳0.1=0.02

By using Baye鈥檚 rule, if both systems sound an alarm, the probability that an intruder is detected obtained as:

PA\B鈱�C=0.855脳0.40.855脳0.4+0.02脳0.6=0.9661

Therefore, the likelihood of spotting an intruder is 0.9661.

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Most popular questions from this chapter

The data for a random sample of six paired observations are shown in the next table.

a. Calculate the difference between each pair of observations by subtracting observation two from observation 1. Use the differences to calculate d炉andsd2.

b. If 渭1补苍诲渭2are the means of populations 1 and 2, respectively, expressed 渭din terms of 渭1补苍诲渭2.

PairSample from Population 1

(Observation 1)

Sample from Population 2(Observation 2)
123456739648417247

c. Form a 95% confidence interval for 渭d.

d. Test the null hypothesis H0:渭d=0against the alternative hypothesis Ha:渭d鈮�0. Use伪=.05 .

Service without a smile. 鈥淪ervice with a smile鈥� is a slogan that many businesses adhere to. However, some jobs (e.g., judges, law enforcement officers, and pollsters) require neutrality when dealing with the public. An organization will typically provide 鈥渄isplay rules鈥� to guide employees on what emotions they should use when interacting with the public. A Journal of Applied Psychology (Vol. 96, 2011) study compared the results of surveys conducted using two different types of display rules: positive (requiring a strong display of positive emotions) and neutral (maintaining neutral emotions at all times). In this designed experiment, 145undergraduate students were randomly assigned to either a positive display rule condition(n1=78)or a neutral display rule condition(n2=67). Each participant was trained to conduct the survey using the display rules. As a manipulation check, the researchers asked each participant to rate, on a scale of 1= 鈥渟trongly agree鈥� to5= 鈥渟trongly disagree,鈥� the statement, 鈥淭his task requires me to be neutral in my expressions.鈥�

a. If the manipulation of the participants was successful, which group should have the larger mean response? Explain.

b. The data for the study (simulated based on information provided in the journal article) are listed in the table above. Access the data and run an analysis to determine if the manipulation was successful. Conduct a test of hypothesis using伪=0.05 .

c. What assumptions, if any, are required for the inference from the test to be valid?

The data is given below

Positive Display Rule:

243333444444444444454444444444444445555555555555555555555555555555555555555555


Neutral Display Rule:

3321211122122232212222212222221222222232122212122322222222222122222

Shopping vehicle and judgment. Refer to the Journal ofMarketing Research (December 2011) study of shopping cart design, Exercise 2.85 (p. 112). Recall that design engineers want to know whether the mean choice of the vice-over-virtue score is higher when a consumer鈥檚 arm is flexed (as when carrying a shopping basket) than when the consumer鈥檚 arm is extended (as when pushing a shopping cart). The average choice score for the n1 = 11 consumers with a flexed arm was x炉1= 59, while the average for the n2 = 11
Consumers with an extended arm was x炉2= 43. In which scenario is the assumption required for a t-test to compare means more likely to be violated, S1= 4 and S2= 2, or, S1= 10 and S2 = 15? Explain.

Question: Refer to the Bulletin of Marine Science (April 2010) study of lobster trap placement, Exercise 6.29 (p. 348). Recall that the variable of interest was the average distance separating traps鈥攃alled trap-spacing鈥攄eployed by teams of fishermen. The trap-spacing measurements (in meters) for a sample of seven teams from the Bahia Tortugas (BT) fishing cooperative are repeated in the table. In addition, trap-spacing measurements for eight teams from the Punta Abreojos (PA) fishing cooperative are listed. For this problem, we are interested in comparing the mean trap-spacing measurements of the two fishing cooperatives.

BT Cooperative

93

99

105

94

82

70

86

PA Cooperative

118

94

106

72

90

66

98


Source: Based on G. G. Chester, 鈥淓xplaining Catch Variation Among Baja California Lobster Fishers Through Spatial Analysis of Trap-Placement Decisions,鈥� Bulletin of Marine Science, Vol. 86, No. 2, April 2010 (Table 1).

a. Identify the target parameter for this study.b. Compute a point estimate of the target parameter.c. What is the problem with using the normal (z) statistic to find a confidence interval for the target parameter?d. Find aconfidence interval for the target parameter.e. Use the interval, part d, to make a statement about the difference in mean trap-spacing measurements of the two fishing cooperatives.f. What conditions must be satisfied for the inference, part e, to be valid?

Question: The purpose of this exercise is to compare the variability of with the variability of .

a. Suppose the first sample is selected from a population with mean and variance . Within what range should the sample mean vary about of the time in repeated samples of measurements from this distribution? That is, construct an interval extending standard deviations of on each side of .

b. Suppose the second sample is selected independently of the first from a second population with mean and variance . Within what range should the sample mean vary about the time in repeated samples of measurements from this distribution? That is, construct an interval extending standard deviations on each side .

c. Now consider the difference between the two sample means . What are the mean and standard deviation of the sampling distribution ?

d. Within what range should the difference in sample means vary about the time in repeated independent samples of measurements each from the two populations?

e. What, in general, can be said about the variability of the difference between independent sample means relative to the variability of the individual sample means?
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