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Magazine Chapter 6: Problem 97
Suppose that \(25 \%\) of the fire alarms in a large city are false alarms. Let
\(x\) denote the number of false alarms in a random sample of 100 alarms.
Approximate the following probabilities:
a. \(P(20 \leq x \leq 30)\)
b. \(P(20
Short Answer
Expert verified
The approximate probabilities are as follows:
a. \(P(20 \leq x \leq 30) \approx 0.7497\)
b. \(P(20 < x < 30) \) is about the same as in (a).
c. \(P(x \geq 35) \approx 0.0104\)
d. The probability that \(x\) is farther than 2 standard deviations from its mean value is approximately \(0.0456\).
Step by step solution
01
Identifying given data
The problem provides us with:
- Probability of false alarms: \(p = 0.25\)
- Random sample size: \(n = 100\)
We have a large random sample of 100 alarms, which allows us to use a binomial distribution to approximate the probabilities.
02
Calculating mean and standard deviation
The mean (\(渭\)) and standard deviation (\(蟽\)) of a binomial distribution are calculated as follows:
Mean: \(渭 = np\)
Standard deviation: \(蟽 = \sqrt{np(1-p)}\)
Using the given data, we can calculate the mean and standard deviation:
Mean: \(渭 = 100 * 0.25 = 25\)
Standard deviation: \(蟽 = \sqrt{100 * 0.25 * (1-0.25)} = \sqrt{18.75} \approx 4.33\)
Now, we can proceed to approximate the probabilities using the z-distribution.
03
Approximating probabilities
To approximate the probabilities, we convert the given ranges to z-scores and look up the probability in a z-table.
The formula for calculating the z-score is: \(z = \frac{x - 渭}{蟽}\)
a. \(P(20 \leq x \leq 30)\)
First, we'll find the z-scores for the range provided:
\(z_{20} = \frac{20-25}{4.33} \approx -1.15\)
\(z_{30} = \frac{30-25}{4.33} \approx 1.15\)
Now, we can find the probability from the z-table: \(P(-1.15 \leq z \leq 1.15) \approx 0.7497\)
b. \(P(20 < x < 30)\)
Using the z-scores calculated above, we can find the probability for \(20 < x < 30\). When comparing less than (<) or greater than (>), we need to subtract one from the z-score (z-1).
So, the probability will be about the same as in (a) since one is small in comparison to the size of the range.
c. \(P(x \geq 35)\)
First, we'll find the z-score for 35:
\(z_{35} = \frac{35-25}{4.33} \approx 2.31\)
Now, we can find the probability from the z-table: \(P(z \geq 2.31) \approx 1 - 0.9896 = 0.0104\)
d. The probability that \(x\) is farther than 2 standard deviations from its mean value.
To find this, we'll first calculate the range:
\(x \leq (渭 - 2蟽) = 25 - 2 * 4.33 = 16.34\)
\(x \geq (渭 + 2蟽) = 25 + 2 * 4.33 = 33.66\)
Now, we'll find the z-scores for the range provided:
\(z_{16.34} = \frac{16.34-25}{4.33} \approx -2\)
\(z_{33.66} = \frac{33.66-25}{4.33} \approx 2\)
Finally, we can find the probability from the z-table:
\(P(x \leq 16.34) = P(z \leq -2) = 0.0228\)
\(P(x \geq 33.66) = P(z \geq 2) = 0.0228\)
Since the probabilities are complementary, we'll add them together:
\(P(x \leq 16.34) + P(x \geq 33.66) = 0.0228 + 0.0228 = 0.0456\)
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
False Alarm Probability
Understanding the probability of false alarms in the context of binomial distribution can often be critical in various fields, such as signal processing or emergency services. In the given exercise, we have a scenario where a city experiences false fire alarms 25% of the time. By defining a random sample of 100 alarms, the exercise challenges us to approximate the likelihood of observing a particular number of false alarms within that sample. Using statistical methods, we can measure this 'false alarm probability' to inform better decision-making and understand potential patterns or deviations in alarm systems.
As we tackle problems like this, it's imperative to realize that binomial distribution provides a framework for predicting the number of successes (in this case, false alarms) based on a known probability of success in a set number of trials. Once we've established these parameters, we can approximate the probabilities for different ranges using normal distribution, giving us valuable insights into the expected behavior of the alarm system.
As we tackle problems like this, it's imperative to realize that binomial distribution provides a framework for predicting the number of successes (in this case, false alarms) based on a known probability of success in a set number of trials. Once we've established these parameters, we can approximate the probabilities for different ranges using normal distribution, giving us valuable insights into the expected behavior of the alarm system.
Standard Deviation
The standard deviation is a measure of how much variance there is from the mean, or expected value. In the context of our false alarm problem, it gives us an idea of how spread out the number of false alarms can be around the average. Once the probability of a false alarm and sample size were provided, we utilized the formula for standard deviation in a binomial distribution: \(\sigma = \sqrt{np(1-p)}\), which boils down to a measure of how much we expect our results (the number of false alarms) to fluctuate.
In the exercise, we determined that the standard deviation was approximately 4.33. This figure is crucial because it sets the stage for us to then use the 'normal approximation to binomial' method to calculate the probability of observing a range of outcomes. When students understand how the standard deviation reflects the variability of the data, interpreting the subsequent calculations in terms of 'how likely' or 'unlikely' becomes much more intuitive.
In the exercise, we determined that the standard deviation was approximately 4.33. This figure is crucial because it sets the stage for us to then use the 'normal approximation to binomial' method to calculate the probability of observing a range of outcomes. When students understand how the standard deviation reflects the variability of the data, interpreting the subsequent calculations in terms of 'how likely' or 'unlikely' becomes much more intuitive.
Normal Approximation to Binomial
The normal approximation to the binomial distribution is an essential tool in statistics, especially when dealing with large sample sizes like in our exercise, which involves 100 alarms. This approximation simplifies calculations by allowing us to use the normal distribution as a representation of the binomial distribution when the sample size is large, and the probability of success is not too close to 0 or 1.
Why is this important? Because the normal distribution has been extensively studied and is well-understood, it has a standardized table (z-table) that makes finding probabilities much faster. But remember, for the normal approximation to be appropriate, certain conditions known as the rules of thumb ought to be met, such as \(np \text{ and } n(1-p)\text{ both being greater than 5 }\). In our fire alarm scenario, we should feel confident using this method for probability approximation due to the substantial sample size (100) and the success probability (0.25) satisfying these conditions.
Why is this important? Because the normal distribution has been extensively studied and is well-understood, it has a standardized table (z-table) that makes finding probabilities much faster. But remember, for the normal approximation to be appropriate, certain conditions known as the rules of thumb ought to be met, such as \(np \text{ and } n(1-p)\text{ both being greater than 5 }\). In our fire alarm scenario, we should feel confident using this method for probability approximation due to the substantial sample size (100) and the success probability (0.25) satisfying these conditions.
Z-Score Calculation
The z-score calculation is a step that transforms our question about false alarms into a question that the standard normal table can answer. By calculating the z-score, we're finding out exactly how many standard deviations away from the mean our value of interest is. The formula \(z = \frac{x - \mu}{\sigma}\) takes any given value 'x' from our original binomial distribution, subtract the mean (\(\mu\)), and divide by the standard deviation (\(\sigma\)). This standardizes the value so it can be compared according to the normal distribution's rules.
In our exercise, we calculated the z-scores for various scenarios, such as the probability of having a number of false alarms within a certain range. By doing this, we could use the z-table to find these probabilities, which are commonly sought after figures in statistical analysis. This conversion to z-scores is a critical step in leveraging the convenience of the normal distribution when analyzing binomial distributions.
In our exercise, we calculated the z-scores for various scenarios, such as the probability of having a number of false alarms within a certain range. By doing this, we could use the z-table to find these probabilities, which are commonly sought after figures in statistical analysis. This conversion to z-scores is a critical step in leveraging the convenience of the normal distribution when analyzing binomial distributions.
