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Chapter 7: Problem 54

Thirty percent of all automobiles undergoing an emissions inspection at a certain inspection station fail the inspection. a. Among 15 randomly selected cars, what is the probability that at most five fail the inspection? b. Among 15 randomly selected cars, what is the probability that between five and 10 (inclusive) fail to pass inspection? c. Among 25 randomly selected cars, what is the mean value of the number that pass inspection, and what is the standard deviation of the number that pass inspection? d. What is the probability that among 25 randomly selected cars, the number that pass is within 1 standard deviation of the mean value?

Short Answer

Expert verified
a. The probability that at most 5 cars fail the inspection can be found through summing individual probabilities calculated with the binomial formula. b. Similarly, we find the probabilities for numbers of failed cars between 5 and 10 (inclusive). c. The mean value of cars passing the inspection is number of cars - (np), and standard deviation is sqrt(n*p*(1-p)) (subtract these from total number of cars). d. The probability that the number of cars passing inspection is within 1 standard deviation from the mean obtained in part c can be found like in part a,b.

Step by step solution

01

Calculating probabilities for part a

The probability that at most 5 cars fail the inspection is the same as the sum of probabilities that 0, 1, 2, 3, 4, 5 cars fail the inspection. Solve each of these using the formula for binomial probability: \(P(X=k) = C(n, k) * (p^k) * ((1-p)^(n-k))\) where \(C(n,k)\) is the combination of n items taken k at a time, n is number of trials (15 in this case), k is number of successes (failures, from 0 to 5) and p is probability of success (.3 in this case). Add those probabilities together.
02

Calculating probabilities for part b

The approach is similar to step 1. This time, the task is to find the probability that between 5 and 10 cars fail the inspection. This means finding and adding up the probabilities for each number of cars failing the inspection from 5 to 10.
03

Calculating mean and standard deviation for part c

For a binomial distribution, the mean is \(np\) and the standard deviation is \(\sqrt{np(1 - p)} \). Here, n is the number of trials(25 here), and p is the failure probability (.3). Subtract the mean from the total number of cars to get the mean of the number of cars that pass, and same for the standard deviation.
04

Calculating probability for part d

Find the probability that the number of cars that pass the inspection is within 1 standard deviation of the mean. This probability is the sum of the probability of the number of cars passing the inspection from (mean - standard deviation) to (mean + standard deviation).

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Key Concepts

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

Emissions Inspection
When an automobile undergoes an emissions inspection, it is checked to ensure that it meets environmental standards for air pollution. If a car emits gases above a certain threshold, it fails the inspection. In this scenario, 30% of cars fail when tested. Emission inspections are crucial for reducing harmful pollutants and ensuring vehicles that operate on the road are environmentally safe.
In emissions inspections, testing a sample of cars helps analysts to understand the failure rate: a crucial aspect of environmental control. By assessing these random samples, we can estimate probabilities and make informed decisions about environmental policies.
Mean and Standard Deviation
The mean and standard deviation are fundamental concepts in probability and statistics. They help describe the distribution of data, particularly in a binomial setup like emissions inspections.
For any binomial distribution, the mean is calculated using the formula:
  • Mean (\overline{X}) = \(np\)
where \(n\)is the total number of trials (e.g., cars tested), and \(p\)is the probability of failure (30% or 0.3 in this example).
The standard deviation is given by:
  • Standard Deviation (\sigma) = \(\sqrt{np(1-p)}\)
This measures the amount of variation in the set of results. Understanding these can help picture how the data is spread around the mean and how likely other results could occur.
Probability Calculation
Probability calculation involves finding the likelihood of different outcomes using mathematical methods. In the context of emissions inspections, we use the binomial probability formula to calculate the chances of a certain number of cars failing or passing.
The formula used is:
  • \( P(X = k) = C(n, k) \cdot (p^k) \cdot ((1-p)^{n-k}) \)
where:
  • \( C(n, k) \)is the number of combinations, or ways, you can have k failures among n trials.
  • \( p \)is the probability of failure.
  • \((1-p)\)is the probability of passing.
This calculation helps us predict various scenarios, such as the probability that no more than five of the cars fail or between five and ten failures occur. By understanding these probabilities, one can make data-driven decisions about car inspections and implement necessary control measures.

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