IE 151 Homework 2 Spring 2014 New Mexico State University

Ie 151 Homework 2 Spring 2014new Mexico State Universitydepartment O

IE 151: Homework 2 Spring 2014 New Mexico State University Department of Industrial Engineering Due Thursday, February 13 th at 11:55PM Section 1: Probability (Show all work in Microsoft Word) – 8 points For this problem consider the following situation. In a bag we have a total of 11 marbles. 4 are red, 3 are blue, 2 are green, and 2 are black. Answer the following questions; use the formulas from probability which we used in class. a) What is the probability of pulling a red marble from the bag? b) What is the probability of pulling either a green or a black marble from the bag? c) How many marbles would you have to pull to guarantee that you have pulled at least one blue marble? d) How much more probable are you to pull a green marble now if you add 2 more green marbles to the bag? e) If the red marbles were half the size of the green marbles, would the probability of pulling a red or a green marble be the same? Defend your answer f) If I add one more of each marble, are the new and the old probabilities equal? Continued on next page… Section 2: Normal Distribution (Show all work in Microsoft Word) – 12 points Little did you know, but Charlie Bucket from Willy Wonka and the Chocolate factory was an Industrial Engineer. The real reason he was selected amongst the other participants was to optimize the way the factory was run. His first project was to optimize the way that the good and bad eggs are determined. Willy has known for some time that unfortunately a number of good eggs have been misclassified as bad eggs. Currently the only way that good and bad eggs are separated is by weight. Charlie’s job is to determine if the weight alone should be the only determining factor when finding a bad egg. The company policy currently states: “Bad eggs come in two sizes, both contain horrible surprises, the good part is missing from one, in the other bad is added for fun.â€� This translates to bad eggs weigh both more and less than good eggs. The following table shows 50 good eggs. Good Eggs Egg Weight Egg Weight Egg Weight Egg Weight Egg Weight 1 39..................................................41 A) What is being measured in this problem B) Is this a continuous or discrete distribution and why? C) Devise a method of randomly sampling from this group of good eggs. Charlie would like work with a sample size of 10 eggs. Describe in detail how Charlie will be selecting the ten eggs for his sample. D) Report the sample that Charlie used E) Sketch this normal distribution accurately using what we know about . Bonus (+2) if you plot using excel. F) Below is a table of 12 bad eggs, plot these eggs on the normal distribution above using red dots. Bad Eggs Egg Weight Egg Weight 1 41............11 G) What is your conclusion about whether or not the weight is enough evidence of a good or bad egg? Use the knowledge we have learned in the class to defend your answer.

Paper For Above instruction

Introduction

The analysis of probability and normal distribution plays a significant role in quality control processes across industries. In this paper, we explore a scenario involving marbles to understand basic probability concepts and then delve into a hypothetical investigation in an egg sorting facility to demonstrate the application of normal distribution principles in quality assessment.

Part 1: Probability with Marbles

The initial problem involves a bag with 11 marbles categorized by color: 4 red, 3 blue, 2 green, and 2 black. The probabilities for various events are analyzed based on these counts.

Probability of Drawing a Red Marble

The probability, P, of drawing a red marble is calculated by dividing the number of red marbles by the total number: P(red) = 4/11 ≈ 0.3636. This straightforward calculation aligns with the classical definition of probability, considering equally likely outcomes.

Probability of Drawing a Green or Black Marble

Since these are mutually exclusive events, the combined probability is the sum of individual probabilities: P(green or black) = P(green) + P(black) = 2/11 + 2/11 = 4/11 ≈ 0.3636.

Number of Draws to Guarantee at Least One Blue Marble

Applying the pigeonhole principle, the worst-case scenario involves drawing all non-blue marbles first. Total non-blue marbles are 8 (4 red + 2 green + 2 black). To ensure at least one blue, you need to draw one more than the number of non-blue marbles: 8 + 1 = 9. Therefore, pulling 9 marbles guarantees at least one blue marble.

Impact of Adding Green Marbles

Adding 2 green marbles increases the total green count to 4, making the total marbles 13. The probability of drawing a green marble becomes P(green) = 4/13 ≈ 0.3077, compared to 2/11 ≈ 0.1818 initially. The increase demonstrates a higher likelihood of drawing green due to the addition.

Size and Probability of Red vs. Green Marbles

If red marbles are half the size of green marbles, the geometric probability based solely on size would differ from the probability based on count, assuming size influences draw probability. Since the probability depends on both size and count, if size is factored into probability weighing (for example, larger size increasing the likelihood), the probabilities would not be the same unless size is disregarded, which contradicts the initial assumption.

Adding One of Each Marble

Adding one marble of each color increases total marbles to 15. The new probabilities are P(red) = 5/15 = 1/3, P(blue) = 4/15, P(green) = 3/15, and P(black) = 3/15. Comparing these to the original probabilities shows they are not equal, indicating that the act of addition changes the probability distribution.

Part 2: Normal Distribution and Egg Classification

Understanding the Measurement

The problem measures egg weight, which is a continuous variable, allowing for an infinite number of possible values within a range. This measurement is crucial for differentiating between good and bad eggs based solely on weight.

Distribution Type Analysis

Since egg weights can take on a continuum of values, the distribution is continuous. The data likely approximate a normal distribution due to the central limit theorem, given the large sample sizes typically involved in quality assessments.

Sampling Method for Good Eggs

Charlie could employ simple random sampling by assigning each egg a unique identifier and then using a random number generator or drawing lots to select 10 eggs. This ensures each egg has an equal chance of selection and prevents bias.

Constructing the Distribution

The sample of 50 good eggs should be used to calculate the mean and standard deviation. A normal distribution curve can then be sketched with these parameters, representing the expected variation in weight.

Plotting with Excel

Plotting the normal distribution involves creating a frequency histogram and superimposing the bell curve using calculated mean and standard deviation. Using Excel functions such as NORM.DIST(), the distribution can be graphically represented.

Bad Egg Data and Distribution Overlay

The 12 bad eggs, with weights close to or deviating from the mean of good eggs, are plotted on the same distribution graph in red dots. This visual comparison helps assess whether weight alone suffices to classify eggs.

Conclusion on Classification Based on Weight

The overlapping weights of good and bad eggs suggest that weight alone is insufficient for accurate classification. The presence of bad eggs within the normal weight range indicates the need for additional methods, such as size, shape, or internal inspection, to improve classification accuracy.

Conclusion

Analyzing probability and distribution is fundamental in manufacturing and quality control. The marble problem illustrates core probability calculations, while the egg classification scenario demonstrates the practical application of normal distribution analysis. A thorough understanding of these concepts enhances decision-making and process optimization across various industries.

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