Problem # 1 (12 points) A scientist is conducting an experiment designed to prod

Question

Problem # 1 (12 points) A scientist is conducting an experiment designed to produce a gene mutation in bacteria. The chance that his experiment will cause a gene mutation in a bacterial colony is 70%. He uses identical treatment on 10 isolated colonies of the bacteria. The scientist considers his results satisfactory if the mutation occurs in at least 8 of the colonies. Let X be a random variable representing the number of bacterial colonies that have mutated. a. Explain why X is a binomial random variable: Specify, in words, what is a trial in this scenario Identify n (the number of trials): Specify. in words, which outcome of a trial would be defined as a "success" Explain why p (is the probability of success) is the same in every trial . dentify p (the probablity of a success).03ee4. (5 points) What is the probability the scientist will consider his results satisfactory? (Use a table or the formula) b. (5 points) c. What is the expected value of mutated bacterial cultures in this sample? (1 point) 10 d. Fill in the blanks in the following sentence: "If the gene mutation experiment was repeated many times, each time using a sample of 20 isolated bacterial cultures, the average number of mutated cultures in a sample would approach ". (1 point)

Explanation / Answer

Ans:

Binomial distribution with n=10,p=0.7

a)Trial is providing gene mutation

number of trials=10

success is causing gene mutation in bacterial colony.

Probability of success,p=0.7 (and it is same as independent trials )

b)P(x=k)=10Ck*0.7k*(1-0.7)10-k

P(satisfactory)=P(x>=8)=P(x=8)+P(x=9)+P(x=10)

=0.2335+0.1211+0.0282=0.3828

c)mean=np=10*0.7=7

d)Now,n changes to 20,so,mean or expectation=np=20*0.7=14

x p(x) 0 0.0000 1 0.0001 2 0.0014 3 0.0090 4 0.0368 5 0.1029 6 0.2001 7 0.2668 8 0.2335 9 0.1211 10 0.0282

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