Question

3. [8 marks] Suppose a survey is conducted by Ipsos, a Canadian market research polling firm, on user satisfaction with cell phone coverage across the country. They sample 10 customers at random without replacement. Assume all sampled customers are independent. Suppose 30% of users nationwide are satisfied with their cell phone coverage.

a) [5 marks] Calculate the probability that 3 or more of the 10 randomly sampled cell phone customers are satisfied with their cell phone coverage.

b) [1 mark] Why is the probability that exactly 3 out of the 10 randomly sampled customers are satisfied with their cell phone coverage different from 0.3? Please answer in at most three sentences.

c) [1 mark] On average, in a sample of 10 customers, how many do you expect to be satisfied with their cell phone coverage?

d) [1 mark] Calculate the variance of the random variable associated with the number of satisfied customers.

Answer 1

Solution:

We are given:

a) Calculate the probability that 3 or more of the 10 randomly sampled cell phone customers are satisfied with their cell phone coverage.

Answer: Let x be the number of cell phone customers that are satisfied with their cell phone coverage. Therefore, we have to find:

We know that:

Therefore, the probability that 3 or more of the 10 randomly sampled cell phone customers are satisfied with their cell phone coverage is 0.6172

b) Why is the probability that exactly 3 out of the 10 randomly sampled customers are satisfied with their cell phone coverage different from 0.3? Please answer in at most three sentences.

Answer: The probability that exactly 3 out of the 10 randomly sampled customers are satisfied with their cell phone coverage is:

  

The probability that exactly 3 out of the 10 randomly sampled customers are satisfied with their cell phone coverage is different from 0.3 because 0.3 is the probability of any randomly selected customer who is satisfied with his/her cell phone coverage and the probability of exactly 3 out of the 10 randomly selected customers is the probability of three customers who are satisfied with their cell phone coverage.

c) On average, in a sample of 10 customers, how many do you expect to be satisfied with their cell phone coverage?

Answer: The expected number of customers in a sample of 10 who are satisfied with their cell phone coverage is:

d) Calculate the variance of the random variable associated with the number of satisfied customers.

Answer: The variance is given below: