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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

Step by step solution

01

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.

02

Calculation for 0.4 success proportion

Repeat Step 1 but now consider p=0.4.

03

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.

04

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability

Probability is the mathematical study that deals with the likelihood of an event occurring. When we talk about probability in the context of a binomial distribution, we consider scenarios where there are two possible outcomes, like success or failure. In our problem about firewall installations, the main event is whether a computer has a firewall installed, representing either success or failure.

The probability of an event is a number between 0 and 1, where 0 means that the event will not happen, and 1 means that the event is certain. For example, if there's an 80% chance (or probability) that a computer owner has installed a firewall, we represent this probability as 0.8.

Probability plays a crucial role in decision-making, such as understanding how likely it is for certain events to happen. Knowing these probabilities helps us make informed predictions and analyses.

Success Proportion

Success proportion, in the context of the exercise, refers to the ratio of successes to total trials. It tells us, on average, how often a certain event occurs. Here, a 'success' means a computer owner has a firewall installed.

When we say the true proportion is 0.8, we mean 80% of computer owners have a firewall. This is our expectation or belief about the general population. Similarly, if the success proportion is 0.4, then 40% of computer owners have a firewall.

• The success proportion is denoted by p in the binomial distribution formula.
• The true success proportion helps us predict outcomes when we survey a small number of people compared to the whole population.
• This concept is useful for comparing predictions to actual survey results to verify assumptions.

Firewall Installation

Firewall installation is an essential security measure for computers to protect personal information from unauthorized access. A firewall acts as a barrier between a trusted network and an untrusted network, like the internet.

In our context, the survey analyzes how many computer owners have actually installed firewalls, compared to the number who claim to have done so. There's a noteworthy difference between reported installations (80%) and actual installations found in a study (40%). This mismatch indicates a need to understand user's actual security practices versus their perceptions.

• Understanding who has a firewall installed helps gauge the effectiveness of cybersecurity awareness.
• Surveys often reveal the discrepancy between self-reported and verified data, prompting further educational efforts.

Survey Analysis

Survey analysis involves collecting and interpreting data from responses to understand trends and beliefs in a population. It often provides insights into people's behaviors and opinions.

In this exercise, survey analysis helps discern the difference between what computer owners claim and what the reality is regarding firewall installations. With surveys, we discover that while many people understand the importance of security (80% claim), only 40% actually implement it.

• Surveys can reveal gaps in awareness versus action, guiding future educational initiatives.
• Analyzing surveys helps companies and organizations assess the effectiveness of their communication and outreach strategies.
• This feedback loop is vital for developing more accurate educational content or security tools.