91Ó°ÊÓ

Let \(X\) be the number of packages being mailed by a randomly selected customer at a certain shipping facility. Suppose the distribution of \(X\) is as follows: \begin{tabular}{l|rrrr} \(x\) & 1 & 2 & 3 & 4 \\ \hline\(p(x)\) & \(.4\) & \(.3\) & \(.2\) & \(.1\) \end{tabular} a. Consider a random sample of size \(n=2\) (two customers), and let \(\bar{X}\) be the sample mean number of packages shipped. Obtain the probability distribution of \(\bar{X}\). b. Refer to part (a) and calculate \(P(\bar{X} \leq 2.5)\). c. Again consider a random sample of size \(n=2\), but now focus on the statistic \(R=\) the sample range (difference between the largest and smallest values in the sample). Obtain the distribution of \(R\). [Hint: Calculate the value of \(R\) for each outcome and use the probabilities from part (a).] d. If a random sample of size \(n=4\) is selected, what is \(P(\bar{X} \leq 1.5)\) ? [Hint: You should not have to list all possible outcomes, only those for which \(\bar{x} \leq 1.5\).]

Step by step solution

01

Find Possible Sample Pairs

List all possible combinations of two customers with their respective number of packages from the distribution. 1. (1, 1) 2. (1, 2) 3. (1, 3) 4. (1, 4) 5. (2, 1) 6. (2, 2) 7. (2, 3) 8. (2, 4) 9. (3, 1) 10. (3, 2) 11. (3, 3) 12. (3, 4) 13. (4, 1) 14. (4, 2) 15. (4, 3) 16. (4, 4)

02

Calculate Sample Means

Calculate the sample mean \( \bar{X} \) for each pair by averaging the values in each pair.1. (1, 1) \( \rightarrow \bar{X} = 1 \) 2. (1, 2) \( \rightarrow \bar{X} = 1.5 \) 3. (1, 3) \( \rightarrow \bar{X} = 2 \) 4. (1, 4) \( \rightarrow \bar{X} = 2.5 \) 5. (2, 1) \( \rightarrow \bar{X} = 1.5 \) 6. (2, 2) \( \rightarrow \bar{X} = 2 \) 7. (2, 3) \( \rightarrow \bar{X} = 2.5 \) 8. (2, 4) \( \rightarrow \bar{X} = 3 \) 9. (3, 1) \( \rightarrow \bar{X} = 2 \) 10. (3, 2) \( \rightarrow \bar{X} = 2.5 \) 11. (3, 3) \( \rightarrow \bar{X} = 3 \) 12. (3, 4) \( \rightarrow \bar{X} = 3.5 \) 13. (4, 1) \( \rightarrow \bar{X} = 2.5 \) 14. (4, 2) \( \rightarrow \bar{X} = 3 \) 15. (4, 3) \( \rightarrow \bar{X} = 3.5 \) 16. (4, 4) \( \rightarrow \bar{X} = 4 \)

03

Determine Probabilities for Sample Means

Find the probability for each unique sample mean by counting the number of occurrences of each mean and calculating the probability using the joint probability rule.- \( \bar{X} = 1 \) with 1 combination: \(0.4 \times 0.4 = 0.16\) - \( \bar{X} = 1.5 \) with 2 combinations: \(2 \times (0.4 \times 0.3) = 0.24\) - \( \bar{X} = 2 \) with 3 combinations: \(3 \times (0.4 \times 0.2) = 0.24\) - \( \bar{X} = 2.5 \) with 4 combinations: \(4 \times (0.4 \times 0.1) = 0.16\) - \( \bar{X} = 2.5 \) with 3 combinations from middle terms: \(3 \times (0.3 \times 0.2) = 0.18\) all accounted for previously- \( \bar{X} = 3 \) with 3 new combinations: \(3 \times (0.2 \times 0.2) = 0.12\) - \( \bar{X} = 3.5 \) with 2 combinations: \(2 \times (0.3 \times 0.1) = 0.06\) - \( \bar{X} = 4 \) with 1 combination: \(0.1 \times 0.1 = 0.01\)

04

Calculate Cumulated Probabilities

Calculate cumulative probability of \( P(\bar{X} \leq 2.5) \) by adding probabilities of all \( \bar{X} \leq 2.5 \):\[ P(\bar{X} = 1) + P(\bar{X} = 1.5) + P(\bar{X} = 2) + P(\bar{X} = 2.5) = 0.16 + 0.24 + 0.24 + 0.34 = 0.98 \].

05

Calculate Sample Range Probabilities

Calculate the range \( R \) for each sample pair and find probabilities.- Range \( R = 0 \) (same numbers) with 4 combinations: \((0.16, 0.09, 0.04, 0.01)\), sum to \(0.30\).- Range \( R = 1 \) with 8 combinations: Calculated similarly, resulting in \(0.40\).- Range \( R = 2 \) with 4 combinations: Leads to \(0.20\). - Range \( R = 3 \) with no new valid combinations: Remains \(0\) as no valid results.

06

Use Hint for New Sample Size

Select only combinations of four customers which give \( \bar{x} \leq 1.5 \), primarily focused on samples with 1s and 2s.- Calculate probabilities similarly as before, only including relevant combinations that have a mean of \(1.5\) or less.To manually verify:- Only sample (1, 1, 1, 1) works: \(0.4^4 = 0.0256\). - Other potential combinations exceed the threshold with smaller numbers of packages.

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Key Concepts

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

Sample Mean

The sample mean is a fundamental concept in statistics, representing the average of a set of observations from a sample. When calculating the sample mean, labeled as \( \bar{X} \), you add up all the observed values and then divide by the number of observations. For instance, in a simple sample of two numbers, (a, b), the sample mean is \( \bar{X} = \frac{a+b}{2} \).

Understanding the sample mean is crucial because it provides a measure of the central tendency of the data. It helps in summarizing large data sets by a single value which serves as an overall representation.

• The sample mean is sensitive to extreme values, or outliers, which means that particularly large or small values can skew the result.
• It is frequently used in probability distributions to understand the average expected outcome.

In the context of the given exercise, calculating the sample mean involved determining the average number of packages shipped by two customers, which required computing the mean for all possible pairs of customers drawn from the given distribution of packages shipped.

Cumulative Probability

Cumulative probability is a probability that the value of a random variable will fall within a specified range or below a specified value. It is an accumulation or sum of probabilities up to a certain point, providing a cumulative view of the probability function.

To calculate it, you sum the probabilities of all outcomes that are less than or equal to the specified value. This concept is particularly useful in understanding the likelihood of events within a certain range rather than focusing on a single outcome.

• In the context of our exercise, the cumulative probability \( P(\bar{X} \leq 2.5) \) was calculated by summing the probabilities of all sample means \( \bar{X} \) less than or equal to 2.5.
• This value helps assess the overall likelihood of shipping smaller numbers of packages on average, which could be important for storage or resource planning in logistics.

Cumulative probabilities are valuable in statistical decision-making, as they give a sense of the total likelihood of varying outcomes occurring below a certain threshold.

Probability Distribution Function

The probability distribution function (PDF) describes how the probabilities are distributed across the possible outcomes of a random variable. For discrete variables, like the number of packages being shipped in this exercise, the probability distribution is often represented in a table form showing each possible value \( x \) along with its probability \( p(x) \).

The PDF provides a complete picture of the random variable's behavior by indicating which outcomes are most likely, and how likely they are.

• In the given exercise, the distribution highlights how likely it is for any number between 1 and 4 packages to be shipped by a customer.
• Such functions are foundational in making predictions and for various calculations in statistics, such as finding expectation values or variances.

Understanding the probability distribution function enables statisticians to gain insights into possible trends and variances within the data, aiding in decision-making processes.

Sample Range

Sample range is a measure of statistical dispersion, indicating the difference between the largest and smallest values in a sample of data. It offers a simple measure of variability within the distribution of data.

To calculate the sample range, denoted as \( R \), you subtract the smallest observation from the largest within the sample range. For example, if you have a sample with values \( a \) and \( b \), the range is \( R = \max(a,b) - \min(a,b) \).

• The range is highly sensitive to outliers because it only considers the two extreme values.
• It can be used to understand the spread or width of the distribution, giving a quick sense of variability, though it doesn't provide information about the distribution between these extremes.

In the exercise, the sample range was used to understand the variation within small samples of customer shipping behavior, indicating how spread out the number of packages being shipped by different customers could be.