91Ó°ÊÓ

Alpha Corporation is considering two suppliers to secure the large amounts of steel rods that it uses. Company A produces rods with a mean diameter of \(8 \mathrm{~mm}\) and a standard deviation of \(15 \mathrm{~mm}\) and sells 10,000 rods for \(\$ 400\). Company B produces rods with a mean diameter of \(8 \mathrm{~mm}\) and a standard deviation of. \(12 \mathrm{~mm}\) and sells 10,000 rods for \(\$ 460\). A rod is usable only if its diameter is between \(7.8 \mathrm{~mm}\) and \(8.2 \mathrm{~mm}\). Assume that the diameters of the rods produced by each company have a normal distribution. Which of the two companies should Alpha Corporation use as a supplier? Justify your answer with appropriate calculations.

Step by step solution

01

Calculate the z-scores for Company A

The z-score is a measure of how many standard deviations an element is from the mean. It is calculated using the formula \(z = \frac{x - \mu}{\sigma}\), where \(x\) is the data point, \(\mu\) is the mean and \(\sigma\) is the standard deviation. In this case, calculate the z-scores for the lower and upper limits of the usable rod diameter for Company A. The mean (\(\mu\)) is 8 mm and the standard deviation (\(\sigma\)) is 0.15 mm. Hence, \(z_1 = \frac{7.8 - 8}{0.15} = -1.33\) and \(z_2 = \frac{8.2 - 8}{0.15} = 1.33\)

02

Calculate the probabilities for Company A

Next, use a z-table (standard normal distribution table) to find the probabilities associated with the calculated z-scores. This table shows how likely it is that a randomly selected data point is less than the z-score. The value corresponding to \(z_1 = -1.33\) is 0.0918, and for \(z_2 = 1.33\) it is 0.9082. The probability of producing a rod within the usable distribution is the difference between these two probabilities, \(P_A = 0.9082 - 0.0918 = 0.8164\). Thus, 81.64% of the rods produced by Company A will be usable.

03

Calculate the z-scores for Company B

Similarly, calculate the z-scores for the lower and upper limits of the usable diameter for Company B, where \(\mu = 8\) mm and \(\sigma = 0.12\) mm. Hence, \(z_1 = \frac{7.8 - 8}{0.12} = -1.67\) and \(z_2 = \frac{8.2 - 8}{0.12} = 1.67\)

04

Calculate the probabilities for Company B

Again, use the z-table. The value corresponding to \(z_1 = -1.67\) is 0.0475, and for \(z_2 = 1.67\) it is 0.9525. The difference, \(P_B = 0.9525 - 0.0475 = 0.9050\), is the probability that Company B produces a rod within the usable range. Thus, 90.50% of the rods produced by Company B will be usable.

05

Compare the probabilities and costs

Although Company B charges more (\$460 vs \$400), it produces a higher percentage (90.50% vs 81.64%) of usable rods. Hence, Company B should be chosen if the higher cost is offset by the reduced waste and increased production efficiency.

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Key Concepts

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

Normal Distribution

In the world of statistics, the normal distribution is a key concept that describes how data is spread out across different values. For Alpha Corporation's steel rods from both suppliers, we assume that the diameters follow this type of distribution.

This is often called a bell curve because it resembles the shape of a bell, where most outcomes fall near the mean. Here's why normal distribution is important:

• Predictability: It allows us to predict the likelihood of certain measurements occurring.
• Symmetry: Most data points cluster around the average, with fewer and fewer occurring as the values move away from the average.
• Standardization: Through tools like Z-scores, we can standardize this information and make meaningful comparisons between different datasets.

Understanding this helps us assess whether a company's production process is consistent and reliable. In this case, both companies A and B have rod diameters that ideally should be normally distributed, with a mean of 8 mm.

Z-score Calculation

The Z-score is a statistical measure that indicates how many standard deviations a data point is from the mean. It helps us understand how unusual or typical a particular value is within a dataset.

For Alpha Corporation's scenario, the Z-score calculation assists in determining whether the rod diameters fall within the usable range (7.8 mm to 8.2 mm). The formula for calculating a Z-score is:\[Z = \frac{x - \mu}{\sigma}\]Where:

• \( x \): The value being compared (e.g., 7.8 mm or 8.2 mm).
• \( \mu \): The mean of the data set (e.g., 8 mm).
• \( \sigma \): The standard deviation of the data set (e.g., 0.15 mm for Company A, and 0.12 mm for Company B).

Calculating Z-scores helps quantify how far a measurement deviates from the mean, enabling precise probability analysis.

Probability Analysis

Probability analysis involves assessing the likelihood of events or outcomes. In this exercise, we want to know the probability that steel rods have diameters within the usable range.

For Company A, with Z-scores of \(-1.33\) and \(1.33\), these scores align with known probabilities from a Z-table. This table displays the cumulative probability up to a given Z-score:

• \(P(z \, \text{<} \, -1.33) \approx 0.0918\) and \(P(z \, \text{<} \, 1.33) \approx 0.9082\)
• The usable range probability for Company A is calculated as \(0.9082 - 0.0918 = 0.8164\), meaning 81.64%.

Similarly for Company B:

• \(P(z \, \text{<} \, -1.67) \approx 0.0475\) and \(P(z \, \text{<} \, 1.67) \approx 0.9525\)
• The probability becomes \(0.9525 - 0.0475 = 0.9050\), resulting in 90.50% usability.

Such analysis informs decision-making about which supplier to use.

Standard Deviation

Standard deviation is a crucial statistical concept that measures the amount of variation or dispersion in a dataset. In our scenario, it tells us how much the diameters of steel rods vary from the mean.

• Small standard deviation: Data points are close to the mean.
• Large standard deviation: Data points are more spread out from the mean.

For Alpha Corporation's choice:

• Company A's standard deviation is 0.15 mm.
• Company B's is 0.12 mm.

A smaller standard deviation, as seen with Company B, indicates more consistency in rod sizes closer to the desired mean, positively influencing the probability of usability.

Mean and Standard Deviation Concepts

The mean is the average value of a dataset, and the standard deviation shows how much variation exists from this average. These two metrics provide essential insights into the data's overall distribution and are elemental in figuring out probability and variability.

• The mean (\(\mu\)) for both suppliers is 8 mm, representing the target rod diameter.
• This uniformity in means reflects the fact that both companies aim for the same average size.
• The different standard deviations (0.15 mm for Company A and 0.12 mm for Company B) illustrate different levels of precision or control within their production processes.

By comparing these values, we assess the reliability of each supplier, deciding which offers a better balance of cost and usability. While both aim for the same mean, the smaller standard deviation of Company B means their product is more consistently closer to this desired value.