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Example 6.14 gave the probability distributions shown below for \(x=\) number of flaws in a randomly selected glass panel from Supplier 1 \(y=\) number of flaws in a randomly selected glass panel from Supplier 2 for two suppliers of glass used in the manufacture of flat screen TVs. If the manufacturer wanted to select a single supplier for glass panels, which of these two suppliers would you recommend? Justify your choice based on consideration of both center and variability. \(\begin{array}{lcccclcccc}\boldsymbol{x} & 0 & 1 & 2 & 3 & \boldsymbol{y} & 0 & 1 & 2 & 3 \\ \boldsymbol{p}(\boldsymbol{x}) & 0.4 & 0.3 & 0.2 & 0.1 & \boldsymbol{p}(\boldsymbol{y}) & 0.2 & 0.6 & 0.2 & 0\end{array}\)

Step by step solution

01

Calculate the mean for both distributions

To calculate the mean (expected value), we will use the formula: \[E(x) = \sum_i{x_i p(x_i)}\] For Supplier 1: \[\mu_x = 0 \cdot 0.4 + 1 \cdot 0.3 + 2 \cdot 0.2 + 3 \cdot 0.1 = 0 + 0.3 + 0.4 + 0.3 = 1\] For Supplier 2: \[\mu_y = 0 \cdot 0.2 + 1 \cdot 0.6 + 2 \cdot 0.2 + 3 \cdot 0 = 0 + 0.6 + 0.4 + 0 = 1\]

02

Calculate the variance for both distributions

To calculate the variance, we will use the formula: \[Var(x) = E(x^2) - E(x)^2\] First, we need to calculate \(E(x^2)\) for both distributions. For Supplier 1: \[E(x^2) = 0^2 \cdot 0.4 + 1^2 \cdot 0.3 + 2^2 \cdot 0.2 + 3^2 \cdot 0.1 = 0 + 0.3 + 0.8 + 0.9 = 2\] For Supplier 2: \[E(y^2) = 0^2 \cdot 0.2 + 1^2 \cdot 0.6 + 2^2 \cdot 0.2 + 3^2 \cdot 0 = 0 + 0.6 + 0.8 + 0 = 1.4\] Now, we can calculate the variances: For Supplier 1: \[Var(x) = E(x^2) - E(x)^2 = 2 - 1^2 = 1\] For Supplier 2: \[Var(y) = E(y^2) - E(y)^2 = 1.4 - 1^2 = 0.4\]

03

Calculate the standard deviation for both distributions

The standard deviation is the square root of the variance. So the standard deviations for both distributions are: For Supplier 1: \[\sigma_x = \sqrt{Var(x)} = \sqrt{1} = 1\] For Supplier 2: \[\sigma_y = \sqrt{Var(y)} = \sqrt{0.4} \approx 0.63\]

04

Compare the center and variability of both distributions

For both suppliers, the mean (center) number of flaws is 1. However, the standard deviation (variability) for Supplier 1 is 1, whereas for Supplier 2 it is approximately 0.63.

05

Make a recommendation based on the comparison

Based on the comparison of both center and variability, we would recommend Supplier 2. The reason is that although both suppliers have the same average number of flaws, the variability in the number of flaws for Supplier 2 is lower (less variation in the quality of glass panels) than Supplier 1. Therefore, Supplier 2 is more consistent and can be considered a better choice.

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Key Concepts

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

Expected Value

The expected value is a fundamental concept in probability theory and statistics used to describe the central tendency of a probability distribution. It is the "center" point of the distribution, or the long-run average of repetitions of the experiment it represents. To find the expected value for a discrete random variable, you multiply each outcome by its probability and then sum these products.

• For Supplier 1: - Use the formula: \[ E(x) = \sum_i{x_i p(x_i)} \] - Calculation: \[ \mu_x = 0 \cdot 0.4 + 1 \cdot 0.3 + 2 \cdot 0.2 + 3 \cdot 0.1 = 1 \]
• For Supplier 2: - Use the same formula: \[ E(y) = \sum_i{y_i p(y_i)} \] - Calculation: \[ \mu_y = 0 \cdot 0.2 + 1 \cdot 0.6 + 2 \cdot 0.2 + 3 \cdot 0 = 1 \]

Both suppliers have an expected value of 1, meaning across many panels, the average number of flaws is predicted to be 1 for both.

Variance

Variance measures the spread of a set of numbers. It helps you understand how much the values in a distribution are dispersed from the expected value. In probability distributions, a high variance indicates that the values are spread out over a wider range, while a low variance indicates that they tend to be closer to the expected value.

• To calculate variance, use: \[ Var(x) = E(x^2) - [E(x)]^2 \]
• Supplier 1's calculations: - First, calculate the second moment: \[ E(x^2) = 0^2 \cdot 0.4 + 1^2 \cdot 0.3 + 2^2 \cdot 0.2 + 3^2 \cdot 0.1 = 2 \] - Then find variance: \[ Var(x) = 2 - 1^2 = 1 \]
• Supplier 2's calculations: - First, calculate the second moment: \[ E(y^2) = 0^2 \cdot 0.2 + 1^2 \cdot 0.6 + 2^2 \cdot 0.2 + 3^2 \cdot 0 = 1.4 \] - Then find variance: \[ Var(y) = 1.4 - 1^2 = 0.4 \]

Supplier 1's variance is higher, meaning more inconsistency in the number of flaws per panel compared to Supplier 2.

Standard Deviation

The standard deviation is a measure that tells us how much the values of a dataset deviate from the mean. It is the square root of the variance and provides a more tangible measure of spread, as it is expressed in the same units as the data.

• Equation: \[ \sigma = \sqrt{Var} \]
• For Supplier 1: - Calculate: \[ \sigma_x = \sqrt{1} = 1 \]
• For Supplier 2: - Calculate: \[ \sigma_y = \sqrt{0.4} \approx 0.63 \]

In this context, Supplier 1's standard deviation is higher, signifying greater variability in flaw count per panel. Supplier 2, with a standard deviation of about 0.63, shows less deviation from the mean, implying more consistency.

Supplier Comparison

Choosing between suppliers should take into account both the expected value and the variability of the number of flaws. While both suppliers have the same expected value of 1, implying an average of one flaw per panel, it's the variability that sets them apart.

• Supplier 1: - Has a variance of 1 and a standard deviation of 1. - This suggests a wider variability in the number of flaws in their glass panels.
• Supplier 2: - Has a lower variance of 0.4 and a standard deviation of roughly 0.63. - This indicates less fluctuation in quality, providing consistency in the manufacturing process.

Therefore, although the average number of flaws is the same, Supplier 2 is preferable as it offers more reliable quality, reducing the likelihood of unexpectedly high numbers of flawed panels.