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Chapter 20: Problem 782

Let \(\mathrm{Y}_{1}<\mathrm{Y}_{2}<\mathrm{Y}_{3}<\ldots \ldots<\mathrm{Y}_{6}\) be the order statistics of a random sample of size six from a continuous distribution. We want to use the interval \(\left(\mathrm{Y}_{1}, \mathrm{Y}_{6}\right)\) as an 80 percent tolerance interval for the distribution. What is the corresponding probability level for this interval?

Short Answer

Expert verified
The probability level for the given 80% tolerance interval, which is the interval \((Y_1, Y_6)\), can be expressed as: \(P(Y_1 < Y < Y_6) = [1 - (1 - F(Y_6))^6] - [F(Y_1)^6]\), where \(F\) is the cumulative distribution function (CDF) of the continuous distribution. This is a non-linear equation, and a closed-form solution may not be possible. However, once a specific continuous distribution is given, we could use numerical methods or properties of the distribution to find an approximate value for the probability level.

Step by step solution

01

Recognize the order statistics

The given order statistics are Y1, Y2, Y3, ..., Y6, which represent the smallest to the largest values in the sample of size six from a continuous distribution.
02

Identify the tolerance interval

We are given an 80 percent tolerance interval, which is the interval (Y1, Y6). This means that we expect at least 80% of the population to fall between the smallest and largest values in our sample.
03

Calculate the probability of an observation falling within the interval

To calculate the probability of an observation falling within the interval (Y1, Y6), we will use the cumulative distribution function (CDF) of the order statistics. The CDF gives the probability that an order statistic is less than or equal to a certain value. Let F be the CDF of the continuous distribution. Then, the probability P(Y1 < Y < Y6) can be expressed as follows: P(Y1 < Y < Y6) = P(Y 鈮� Y6) - P(Y 鈮� Y1)
04

Express the CDF in terms of the order statistics

The CDF can be expressed in terms of the order statistics Y1 and Y6 as follows: P(Y 鈮� Y6) = [1 - (1 - F(Y6))^6] P(Y 鈮� Y1) = [F(Y1)^6]
05

Combine the expressions for the probability level

Using the expressions from Step 4, we can now express the probability level P(Y1 < Y < Y6) as follows: P(Y1 < Y < Y6) = [1 - (1 - F(Y6))^6] - [F(Y1)^6] Let p = P(Y1 < Y < Y6) Now, we are given that 80% of the population falls within the interval (Y1, Y6): 0.8 = 1 - (1 - F(Y6))^6 - F(Y1)^6
06

Solve for the probability level

Now, we have an equation involving the CDF and the probability level, which we need to solve for p: 0.8 = 1 - (1 - F(Y6))^6 - F(Y1)^6 Since the equation is in terms of order statistics and involves a continuous distribution, it may not be possible to find a closed form solution for the probability level. However, we can infer that this is a non-linear equation and may require numerical methods for obtaining an approximate solution. At any rate, we have derived the equation that will help us find the probability level for the given 80% tolerance interval. Once a specific continuous distribution is given, we could use numerical methods or properties of the distribution to find an approximate value for the probability level.

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Key Concepts

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

Tolerance Interval
A tolerance interval is a statistical interval within which we expect a certain percentage of a population to fall. It differs from confidence intervals or prediction intervals in that it specifically reflects the spread of the entire population, not just the sample or future observations. In our exercise, we have an 80% tolerance interval for the distribution of order statistics from a continuous distribution.
This means that we anticipate 80% of all possible data values from this distribution to lie within the calculated interval.
  • Key Point: A tolerance interval provides a range that captures a specified portion of a population with a certain degree of confidence.
  • Application: In quality control and risk management, tolerance intervals help in defining boundaries that conform to required reliability standards.
  • Usage: Engineers might use tolerance intervals to determine if manufactured parts meet specifications when variability is an element.
Continuous Distribution
A continuous distribution is a type of probability distribution that assumes an infinite number of possible values within a given range. Because of this, we use probability density functions and cumulative distribution functions (CDF) to understand their behavior. In the context of order statistics, we're dealing with continuous distributions to represent the underlying data of our sample.
This distribution aids in calculating probabilities and setting intervals with regard to the entire data range.
  • Nature of Continuous Distributions: Values are spread over an interval instead of discrete points.
  • Examples: Normal distribution, exponential distribution, and uniform distribution are common forms.
  • Importance: Modeling phenomena in the real world, such as heights, weights, or test scores, typically assumes continuous distributions.
Cumulative Distribution Function
The cumulative distribution function, or CDF, provides the probability that a random variable is less than or equal to a certain value. It's crucial for understanding how probabilities accumulate in a distribution. In our original problem, the CDF helps in evaluating order statistics, where calculating the distribution's range is necessary.
The CDF has several key properties:
  • Range: It always ranges from 0 to 1, increasing with the value of the random variable.
  • Utility: By determining P(Y 鈮� y), it helps in setting boundaries and intervals for probability calculations.
  • Link with Tolerance Intervals: In our context, using the CDF of order statistics allows us to determine the percentage of data captured within a tolerance interval.
For example, with symmetric distributions, understanding the CDF can directly inform which percentile points you consider, like the minimum or maximum values as seen in the interval (Y1, Y6). This function becomes especially useful when computing probabilities related to order statistics, where specific numerical points are crucial for calculations.

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Most popular questions from this chapter

The following table gives scores on equivalent forms of the same test taken before \(\left(\mathrm{A}_{1}\right)\) and after \(\left(\mathrm{A}_{2}\right)\) a special audiovisual instructional program. Determine whether these data support the conclusion that this special program improved test performance. $$ \begin{array}{|l|l|l|l|l|l|} \hline \text { Person } & \mathrm{A}_{1} & \mathrm{~A}_{2} & \text { Person } & \mathrm{A}_{1} & \mathrm{~A}_{2} \\ \hline 1 & 87 & 98 & 12 & 72 & 79 \\ \hline 2 & 89 & 93 & 13 & 80 & 78 \\ \hline 3 & 83 & 90 & 14 & 73 & 77 \\ \hline 4 & 93 & 89 & 15 & 77 & 76 \\ \hline 5 & 85 & 87 & 16 & 70 & 75 \\ \hline 6 & 81 & 86 & 17 & 66 & 73 \\ \hline 7 & 82 & 85 & 18 & 69 & 71 \\ \hline 8 & 78 & 84 & 19 & 71 & 70 \\ \hline 9 & 79 & 83 & 20 & 58 & 67 \\ \hline 10 & 75 & 81 & 21 & 64 & 61 \\ \hline 11 & 74 & 80 & & & \\ \hline \end{array} $$

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