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A shipping company handles containers in three different sizes: (1) 27 ft³ (3 × 3 × 3), (2) 125 ft³, and (3) 512 ft³. Let X; (i = 1, 2, 3) denote the number of type i containers shipped during a given week. With μ; = E(X;) and σ;² = V(X;), suppose that th mean values and standard deviations are as follows: M₁ = 200 %₁ = 11 = 260 H₂ %₂ = 13 H3 = 110 03 = 7 (a) Assuming that X₁, X₂, X3 are independent, calculate the expected value and variance of the total volume shipped. [Hin Volume = 27X₁ + 125X₂ + 512X3.] expected value ft3 variance ft6 (b) Would your calculations necessarily be correct if the X;'s were not independent? Explain. O Both the expected value and the variance would be correct. O The expected value would not be correct, but the variance would be correct. O Neither the expected value nor the variance would be correct. The onog o would not be correct

A shipping company handles containers in three different sizes: (1) 27 ft³ (3 × 3 × 3), (2) 125 ft³, and (3) 512 ft³. Let X; (i = 1, 2, 3) denote the number of type i containers shipped during a given week. With μ; = E(X;) and σ;² = V(X;), suppose that th mean values and standard deviations are as follows: M₁ = 200 %₁ = 11 = 260 H₂ %₂ = 13 H3 = 110 03 = 7 (a) Assuming that X₁, X₂, X3 are independent, calculate the expected value and variance of the total volume shipped. [Hin Volume = 27X₁ + 125X₂ + 512X3.] expected value ft3 variance ft6 (b) Would your calculations necessarily be correct if the X;'s were not independent? Explain. O Both the expected value and the variance would be correct. O The expected value would not be correct, but the variance would be correct. O Neither the expected value nor the variance would be correct. The onog o would not be correct

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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**Title: Shipping Company Container Management: Expected Value and Variance Calculation** --- **Summary:** A shipping company ships containers in three different sizes: 1. 27 ft³ (3 × 3 × 3) 2. 125 ft³ 3. 512 ft³ Let \( X_i \) (where \( i = 1, 2, 3 \)) denote the number of type \( i \) containers shipped in a given week. Suppose the mean values and standard deviations are as follows: - \( \mu_1 = 200 \), \( \sigma_1 = 11 \) - \( \mu_2 = 260 \), \( \sigma_2 = 13 \) - \( \mu_3 = 110 \), \( \sigma_3 = 7 \) **Task 1: Calculate Expected Value and Variance** (a) Assuming that \( X_1, X_2, X_3 \) are independent, calculate the expected value and variance of the total volume shipped. **Hint:** Volume = \( 27X_1 + 125X_2 + 512X_3 \). - **Expected Value:** ______ ft³ - **Variance:** ______ ft⁶ **Task 2: Independence Impact on Calculations** (b) Would your calculations necessarily be correct if the \( X_i \)'s were not independent? Explain. - Options: - Both the expected value and the variance would be correct. - The expected value would not be correct, but the variance would be correct. - Neither the expected value nor the variance would be correct. - The expected value would be correct, but the variance would not be correct. --- **Educational Explanation:** This question involves statistical concepts such as expected values and variance, which are key in predicting and understanding variability in shipping volumes. When components are treated as independent, calculations are often simpler. This exercise encourages thinking about the implications of dependency between variables, which can complicate variance calculations, providing valuable insight into real-world applications.
Transcribed Image Text:**Title: Shipping Company Container Management: Expected Value and Variance Calculation** --- **Summary:** A shipping company ships containers in three different sizes: 1. 27 ft³ (3 × 3 × 3) 2. 125 ft³ 3. 512 ft³ Let \( X_i \) (where \( i = 1, 2, 3 \)) denote the number of type \( i \) containers shipped in a given week. Suppose the mean values and standard deviations are as follows: - \( \mu_1 = 200 \), \( \sigma_1 = 11 \) - \( \mu_2 = 260 \), \( \sigma_2 = 13 \) - \( \mu_3 = 110 \), \( \sigma_3 = 7 \) **Task 1: Calculate Expected Value and Variance** (a) Assuming that \( X_1, X_2, X_3 \) are independent, calculate the expected value and variance of the total volume shipped. **Hint:** Volume = \( 27X_1 + 125X_2 + 512X_3 \). - **Expected Value:** ______ ft³ - **Variance:** ______ ft⁶ **Task 2: Independence Impact on Calculations** (b) Would your calculations necessarily be correct if the \( X_i \)'s were not independent? Explain. - Options: - Both the expected value and the variance would be correct. - The expected value would not be correct, but the variance would be correct. - Neither the expected value nor the variance would be correct. - The expected value would be correct, but the variance would not be correct. --- **Educational Explanation:** This question involves statistical concepts such as expected values and variance, which are key in predicting and understanding variability in shipping volumes. When components are treated as independent, calculations are often simpler. This exercise encourages thinking about the implications of dependency between variables, which can complicate variance calculations, providing valuable insight into real-world applications.
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