91Ó°ÊÓ

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 3000 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 .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 .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 have a firewall installed is \(.40\) or \(.80\) ? Justify your answer based on probability calculations.

Step by step solution

01

Calculate Probability with .80 Success Rate

We first calculate the probability that more than 15 out of 20 computer owners have a firewall installed with a .80 success rate. This can be done using the complement rule of probability. The probability of more than 15 successes is the complement of the probability of 15 or fewer successes. We use the binomial probability formula: \(P(X > 15) = 1 - P(X \leq 15)\). Using BINOM.DIST function in excel, we get \(P(X \leq 15) = 0.558\), hence the result is \(P(X > 15) = 1 - 0.558 = 0.442 \)

02

Calculate Probability with .40 Success Rate

Next, we calculate the probability that more than 15 out of 20 computer owners have a firewall installed with a .40 success rate. Similarly, we use the complement rule: \(P(X > 15) = 1 - P(X \leq 15)\). Using BINOM.DIST function in excel, we get \(P(X \leq 15) = 1\), hence the result is \(P(X > 15) = 1 - 1 = 0 \)

03

Compare Probabilities for Part C

Finally, we calculate the probabilities of getting exactly 14 out of 20 computer owners having a firewall installed with both the .80 and .40 success rate. Then we compare those two probabilities to determine which is more likely. Using BINOM.DIST function in excel, we get \(P(14; .80) = 0.210\) and \(P(14; .40) = 0.250\). So, it's more likely to get exactly 14 successes with the .40 success rate.

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Key Concepts

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

Probability Calculation

Understanding probability calculation is essential when analyzing scenarios like the firewall installation discussed in the exercise. Probability is a measure of the likelihood that a particular event will occur, and is expressed as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty.

In the context of the exercise, calculating the probability involves determining how likely it is for a certain number of computer owners out of a sample to have a firewall installed. The basic formula to compute the probability of an event is the ratio of the number of favorable outcomes to the total number of possible outcomes. However, for more complex scenarios involving multiple trials, such as those following a binomial distribution, specialized formulas and possible use of technology like Excel or calculators come into play.

Binomial Distribution

The binomial distribution is a probability distribution that summarizes the likelihood that a value will take on one of two independent values under a given set of parameters or assumptions. Specifically, it models the number of successes in a fixed number of independent trials, with the same probability of success in each trial.

In the exercise, we're dealing with binomial distribution since the event of a computer owner having a firewall can be considered a success, and not having one as a failure. The trials are random selections of computer owners, and each trial (owner) is independent of others. The formula for the binomial probability of observing exactly 'k' successes in 'n' trials is given by: \( P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \), where \(p\) is the probability of success on each trial, \(n\) is the total number of trials, and \(k\) is the number of successes we want to calculate the probability for. In the exercise's scenario, the BINOM.DIST function in Excel makes the computation practical and quick.

Complement Rule

The complement rule is a foundation in probability theory. It states that the sum of the probabilities of an event and its complement is 1. This is because the complement of an event includes all possible outcomes that are not part of the original event.

For instance, if you want to find the probability that more than 15 computer owners have a firewall, you first find the probability of the opposite - 15 or fewer owners having it. Once you have that probability, the complement rule allows you to subtract this from 1 to get the desired probability. Mathematically it is expressed as: \(P(A^c) = 1 - P(A)\). In our exercise, this rule simplifies the process since calculating the probability of having more than 15 successes directly would be more complex.

Statistical Surveys

Statistical surveys are a systematic method for collecting data from a specific population to draw conclusions about that population. The survey mentioned in the textbook problem illustrates how data collected from a sample can help estimate computer security measures among computer owners.

However, surveys must be carefully designed to avoid biases and inaccuracies. The selection of participants should be random and representative of the population, and the questions should be clear and objective. Statistical surveys can be powerful tools in data-driven decision-making, but their reliability is dependent on the methodology and the interpretation of their results. Regarding the firewall example, the discrepancy between the claimed and observed firewall installations points to potential issues in the survey process or participant answers, highlighting the importance of careful survey design and execution.