According to a 2014 study by TSYS, 30% of shoppers prefer to pay for online purc

Question

According to a 2014 study by TSYS, 30% of shoppers prefer to pay for online purchases with a debit card. Consider a group of 20 shoppers who were randomly selected for a focus group. The random variable X represents the number of people in the focus group who prefer to pay for online purchases with a debit card. What is the probability that exactly 8 people in the focus group prefer to pay for online purchases with a debit card? What is the probability that 4 or more people in the focus group prefer to pay for online purchases with a debit card? What is the probability that fewer than 3 people in the focus group prefer to pay for online purchases with a debit card? What is the expected number of people in the focus group who prefer to pay for online purchases with a debit card? What is the variance of the number of people in the focus group who prefer to pay for online purchases with a debit card?

Explanation / Answer

Solution:-

a) The probability that exactly 8 people in the focus group prefer to pay online purchases with debit card is 0.1144.

p = 30/100 = 0.30

n = 20

x = 8

By applying Binomial distribution:-

P(x, p, n) = nCx*px*(1 - p)(n - x)

P(x = 8) = 0.1144

b) The probability that 4 or more people in the focus group prefer to pay online purchases with debit card is 0.8929  

x = 4

By applying Binomial distribution:-

P(x, p, n) = nCx*px*(1 - p)(n - x)

P(x > 4) = 0.8929

c) The probability that fewer than 3 people in the focus group prefer to pay online purchases with debit card is 0.03548.

x = 3

By applying Binomial distribution:-

P(x, p, n) = nCx*px*(1 - p)(n - x)

P(x < 3) = 0.03548

d) The expected number of people in the focus group who prefer to pay for online purchases with a debit card is 6.0.

E(x) = n × p

E(x) = 20 × 0.30

E(x) = 6.0

e) The variance of the number of people in the focus group who prefer to pay for online purchases with a debit card is 4.2.

Varince = n × p × (1 - p)

Variance = 20 × 0.30 × 0.70

Variance = 4.2

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