At the beginning of October of 2020, based upon the results of over 1.5 million antibody tests, it was determined that 27% of New York City residents had COVID-19 antibodies. Suppose we took a random sample of n=19 New York City residents at the beginning of October. Let X = the number of these selected residents who have COVID-19 antibodies. (Do not use a normal approximation for this problem. This is a binomial problem.) a) What is the probability that X < 6?  b) What is the probability that 5 ≤ X ≤ 7?  c) What is the probability that X = 6?

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At the beginning of October of 2020, based upon the results of over 1.5 million antibody tests, it was determined that 27% of New York City residents had COVID-19 antibodies. Suppose we took a random sample of n=19 New York City residents at the beginning of October. Let X = the number of these selected residents who have COVID-19 antibodies. (Do not use a normal approximation for this problem. This is a binomial problem.)

a) What is the probability that X < 6? 

b) What is the probability that 5 ≤ X ≤ 7? 

c) What is the probability that X = 6? 

d) What is the expected value of X? 

e) What is the variance of X?

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