91Ó°ÊÓ

Let \(X_{1}, X_{2}, \ldots, X_{n}\) be a random sample from a continuous probability distribution having median \(\tilde{\mu}\) (so that \(\left.P\left(X_{l} \leq \tilde{\mu}\right)=P\left(X_{l} \geq \tilde{\mu}\right)=.5\right)\). a. Show that $$ P\left(\min \left(X_{i}\right)<\tilde{\mu}<\max \left(X_{j}\right)\right)=1-\left(\frac{1}{2}\right)^{n-1} $$ so that \(\left(\min \left(x_{i}\right), \max \left(x_{i}\right)\right)\) is a \(100(1-\alpha) \%\) confidence interval for \(\tilde{\mu}\) with \(\alpha=\left(\frac{1}{2}\right)^{n-1}\). [Hint: The complement of the event \(\left\\{\min \left(X_{j}\right)<\widetilde{\mu}<\max \left(X_{j}\right)\right\\}\) is \(\left\\{\max \left(X_{j}\right) \leq\right.\) \(\tilde{\mu}\\} \cup\left\\{\min \left(X_{i}\right) \geq \tilde{\mu}\right\\}\). But \(\max \left(X_{i}\right) \leq \tilde{\mu}\) iff \(X_{i} \leq \tilde{\mu}\) for all \(i .]\) b. For each of six normal male infants, the amount of the amino acid alanine \((\mathrm{mg} / 100 \mathrm{~mL}\) ) was determined while the infants were on an isoleucine-free diet, resulting in the following data: \(\begin{array}{llllll}2.84 & 3.54 & 2.80 & 1.44 & 2.94 & 2.70\end{array}\) Compute a \(97 \%\) CI for the true median amount of alanine for infants on such a diet ("The Essential Amino Acid Requirements of Infants," Amer. \(J\). of Nutrition, 1964: 322-330). c. Let \(x_{(2)}\) denote the second smallest of the \(x_{i}\) 's and \(x_{(1 n-1)}\) denote the second largest of the \(x_{i}\) 's. What is the confidence level of the interval \(\left(x_{(2)}, x_{(n-1)}\right)\) for \(\tilde{\mu}\) ?

Step by step solution

01

Understanding the Complements

The event we're interested in is \(\min \left(X_{i}\right)<\tilde{\mu}<\max \left(X_{j}\right)\). The complement of this event happens when \(\max \left(X_{i}\right) \leq \tilde{\mu}\) or \(\min \left(X_{i}\right) \geq \tilde{\mu}\). If the maximum value is less than or equal to \(\tilde{\mu}\), then all \(X_i\) are less than or equal to \(\tilde{\mu}\). If the minimum value is greater than or equal to \(\tilde{\mu}\), then all \(X_i\) are greater than or equal to \(\tilde{\mu}\).

02

Calculating Complement Probabilities

The probability that all \(X_i \leq \tilde{\mu}\) is \((0.5)^n\) (since each \(X_i\) independently has a 0.5 probability of being less than \(\tilde{\mu}\)), and similarly, the probability that all \(X_i \geq \tilde{\mu}\) is also \((0.5)^n\). Therefore, the probability of the complement event is the sum of these probabilities: \(2 \times (0.5)^n = 2^{1-n}\).

03

Determining the Confidence Interval

The probability of the event where \(\tilde{\mu}\) is between the minimum and maximum is the complement of the above event. Thus, \(P(\min (X_i) < \tilde{\mu} < \max (X_j)) = 1 - 2^{1-n}\). For the confidence interval, \(1 - \alpha = 1 - \left(\frac{1}{2}\right)^{n-1}\), and so \(\alpha = \left(\frac{1}{2}\right)^{n-1}\).

04

Analyzing the Alanine Data

Given data are: \(2.84, 3.54, 2.80, 1.44, 2.94, 2.70\). Sorting these values gives: \(1.44, 2.70, 2.80, 2.84, 2.94, 3.54\). The smallest value is 1.44 and the largest is 3.54.

05

Compute 97% Confidence Interval

For a 97% confidence interval, \(\alpha = 0.03\) and since \(2^{1-6} = \frac{1}{32}\approx 0.03125\), it aligns with a confidence level very close (97%) for this six-sample size. Therefore, the interval is \((1.44, 3.54)\).

06

Personnel Confidence Level of the Interval (x(2), x(n-1))

In the sample \(2.70, 2.80, 2.84, 2.94\), \(x_{(2)}\) is 2.70 and \(x_{(n-1)}\) is 2.94. The confidence level is found by removing only one extreme value from each end, thus the interval is slightly less wide. From previous complement calculations, the confidence level decreases to \(1 - 2 \times \left(\frac{1}{2}\right)^{4} = \frac{14}{16} = 0.875\%\), or 87.5%.

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Key Concepts

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

Continuous Probability Distribution

A continuous probability distribution is a fundamental concept in probability theory. It represents outcomes that can take any value within a given range. Unlike discrete distributions, which involve distinct, separate values, continuous distributions involve a continuum of possible outcomes.
One of the key aspects of continuous probability distributions is that the probability of an exact single outcome is zero. Instead, probabilities are determined over an interval of values. Common examples include the normal distribution and exponential distribution.
Understanding continuous probability distributions is crucial because they provide a foundation for statistical analysis. Many statistical methods, such as hypothesis testing and confidence intervals, rely on these distributions to make inferences about population parameters.

Median

The median is a central concept in statistics, representing the middle value in a sorted dataset. It's particularly useful because, unlike the mean, it is not skewed by extreme values, providing a better measure of central tendency for skewed distributions. In simpler terms, it divides a data set into two equal halves.
To find the median:

• Sort the data from smallest to largest.
• If the number of data points is odd, the median is the middle number.
• If the number of data points is even, the median is the average of the two middle numbers.

For a probability distribution, the median is the value that separates the higher half from the lower half. In a continuous probability distribution, the median is the point where the cumulative distribution function (CDF) is 0.5, meaning there's a 50% chance of observing a value below the median and a 50% chance above.

Complementary Event Probability

Complementary event probability is a concept in probability that deals with the likelihood of an event not occurring. If 'A' is an event, then the complement of 'A' (often denoted as 'A complement') is the set of all outcomes not in 'A'.
The probability of an event and its complement always adds up to 1:\[ P(A) + P(A') = 1 \]This concept is particularly useful in confidence interval calculations where understanding the distribution and the probability of extreme, non-typical events is important.
In the context of our exercise, the complement helps decide how likely it is for the median to lie within a particular range of the dataset. By establishing this, we can understand the reliability of our confidence interval in capturing the median within those bounds.

Alanine Measurement

Alanine measurement refers to determining the concentration of the amino acid alanine in a given sample. Alanine is an amino acid involved in various metabolic processes, and its measurement is often important in nutritional studies and clinical research.
In our exercise, the determination of alanine concentration in infants on a specific diet helps researchers understand the dietary impact on nutrient levels. Measuring alanine accurately can provide insights into metabolic conditions or the effectiveness of dietary interventions.
The measurement involves collecting samples, such as blood or serum, and using analytical methods to quantify the alanine level, usually expressed in milligrams per 100 milliliters (mg/100 mL). This standardization allows comparing results across different studies and understanding deviations or changes due to specific conditions.