Question

In a press release dated October 2,2008 , The National Cyber Security Alliance reported that approximately \(80 \%\) of adult Americans who own a computer claim to have a firewall installed on their computer to prevent hackers from stealing personal information. This estimate was based on a survey of 3,000 people. It was also reported that in a study of 400 computers, only about \(40 \%\) actually had a firewall installed. a. Suppose that the true proportion of computer owners who have a firewall installed is \(0.80 .\) If 20 computer owners are selected at random, what is the probability that more than 15 have a firewall installed? b. Suppose that the true proportion of computer owners who have a firewall installed is \(0.40 .\) If 20 computer owners are selected at random, what is the probability that more than 15 have a firewall installed? c. Suppose that a random sample of 20 computer owners is selected and that 14 have a firewall installed. Is it more likely that the true proportion of computer owners who

Short answer

The probability of more than 15 out of 20 computer owners having a firewall installed would be different under assumption of success rates of 0.8 and 0.4. After comparing the probabilities of 14 successes under both success rates, we can infer which success rate is more plausible for the observed outcome.

Step-by-step solution

Calculation for 0.8 success proportion

Calculate the probability of more than 15 out of 20 having a firewall. This is done by summing up the probabilities of exactly 16, 17, 18, 19, 20 successes. We use the binomial distribution formula: \[ P(x=k) = C(n, k) * (p^k) * ((1-p)^(n-k)) \] where \( C(n, k) = \frac{n!}{k!(n-k)!} \) is the binomial coefficient, n is the number of trials, k is the number of successes and p is the probability of success. Here, n=20 and p=0.8.

Calculation for 0.4 success proportion

Repeat Step 1 but now consider p=0.4.

Checking observed outcome

Calculate the probabilities P(x=14) for both success rates, 0.8 and 0.4. Follow the formula set out in Step 1.

Decision

Compare the calculated values. It's more likely that the true proportion of computer owners who have a firewall installed is closest to the success proportion having the highest probability for the observed outcome of 14 successes.