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Please explain in the problem below how a) λ was obtained and b) what went wrong with the Poisson approximation.

Please explain in the problem below how a) λ was obtained and b) what went wrong with the Poisson approximation.

College Algebra
College Algebra
10th Edition
ISBN:9781337282291
Author:Ron Larson
Publisher:Ron Larson
Chapter8: Sequences, Series,and Probability
Section8.7: Probability
Problem 50E: Flexible Work Hours In a recent survey, people were asked whether they would prefer to work flexible...
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Please explain in the problem below how a) λ was obtained and b) what went wrong with the Poisson approximation.

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Example 5.8. It is known that of 313.6 million of US population, about 200 million have high-speed Internet access at home. Thus the population proportion of high-speed Internet users is 0.6378. Suppose 1500 people are randomly selected. What is the probability that at least 1000 of those responding to the survey have high-speed Internet access? Solution. We interpret 0.6378 proportion as p and a sample of 1500 people selected for the survey as a sample of 1500 independent Bernoulli trials, so that X = X₁ + ... + X1500 € [B(1500, 0.6378)] gives the number of those in the surveyed sample who have high-speed Internet access. Thus, we are interested to find
Transcribed Image Text:Example 5.8. It is known that of 313.6 million of US population, about 200 million have high-speed Internet access at home. Thus the population proportion of high-speed Internet users is 0.6378. Suppose 1500 people are randomly selected. What is the probability that at least 1000 of those responding to the survey have high-speed Internet access? Solution. We interpret 0.6378 proportion as p and a sample of 1500 people selected for the survey as a sample of 1500 independent Bernoulli trials, so that X = X₁ + ... + X1500 € [B(1500, 0.6378)] gives the number of those in the surveyed sample who have high-speed Internet access. Thus, we are interested to find
1200) 0.6378 0.36221500-k Now the above is a computational challenge. The normal approximation, due to the Central Limit Theorem (to be explored in Chapter V, section 2), would be a remedy. Alternatively, we can use R language to compute the above probability precisely. The procedure will be as follows. P{X > 1000} = 1− P{X < 1000} = 1 − P{X ≤ 999} P{X ≥ 1000} = 150 implying k=1000 > 1-pbinom (999,1500,.6378) [1] 0.01042895 which reads P{X > 1000} = 0.01042895. Here pbinom (999,1500,.6378) calculates 999 Σ (¹500) 0.6378* 0.3622¹500-k Now if we use the Poisson approximation with λ = 956.7 (why?) we arrive at > 1-ppois(999,956.7) [1] 0.08395288 The result significantly differs from that of direct binomial. The student needs to explain the discrepancy. (See Problem 5.6.)
Transcribed Image Text:1200) 0.6378 0.36221500-k Now the above is a computational challenge. The normal approximation, due to the Central Limit Theorem (to be explored in Chapter V, section 2), would be a remedy. Alternatively, we can use R language to compute the above probability precisely. The procedure will be as follows. P{X > 1000} = 1− P{X < 1000} = 1 − P{X ≤ 999} P{X ≥ 1000} = 150 implying k=1000 > 1-pbinom (999,1500,.6378) [1] 0.01042895 which reads P{X > 1000} = 0.01042895. Here pbinom (999,1500,.6378) calculates 999 Σ (¹500) 0.6378* 0.3622¹500-k Now if we use the Poisson approximation with λ = 956.7 (why?) we arrive at > 1-ppois(999,956.7) [1] 0.08395288 The result significantly differs from that of direct binomial. The student needs to explain the discrepancy. (See Problem 5.6.)
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